Some of them have backings, but others don't. Most of the characters are based on characters from Polynesian myths, but the details in some cases do not line up with the mythology.
Let's take a look...
Note: There are a lot of versions of Māui, from different islands. I'm going to lump them all together - I can't separate them, as not all sources state which island they're from.
Māui has tattoos all over his body in the Moana film. While I can't find an explicit reference to Māui having tattoos, tattoos were an important part of Polynesian culture and showed things like your accomplishments - just like Māui's show his (but without moving).
Māui in the movie says how his parents didn't like him, and decided to get rid of him:
Māui: I wasn't born a demigod. I have human parents. They... They took one look. And decided they did not want me. They threw me under the sea. Like I was nothing. Somehow I was found by the Gods. They gave me the hook. They made me... Māui.
This is borne out by the myths. He was left abandoned, and found by the gods, and taken up to the Realm of the Gods.
For this one, we're going to need go through the song "You're Welcome":
Let's go through each claim individually:
What can I say except "you're welcome"?
For the tides, the sun, the sky
Hey, it's okay, it's okay, you're welcome
So, we have a claim that he created the tides, the sun, and the sky:
1.) The tides. This appears to be FALSE - Māui does not have anything to do with the tides. The tides are the responsibility of Tangaroa  .
2.) The sun. He's not fully responsible for the sun - that appears to be (one of) the job(s) of the god known as Kāne . However, he does have something to do with the sun, so half-credit - see later on in the song.
3.) The sky. Again, this seems to not be exclusively Māui's job , but he does have stuff to do with the sky - again, see later, where he goes into more detail.
Hey, what has two thumbs and pulled up the sky
When you were waddling yay high? This guy!
4.) The sky (2). This seems to be accurate. There are a couple of different versions of why he did this, though. Some say that he did it because he saw that humans were suffering. Others    say he did it to impress a girl. There are also vastly different versions of how he did this. Regardless, this appears to be accurate.
When the nights got cold, who stole you fire from down below?
You're looking at him, yo!
Again, this appears to be accurate - it's one of the most popular Māui myths that I can find! Accounts vary, but this is a genuine Polynesian myth  .
Oh, also I lassoed the sun, you're welcome
To stretch your days and bring you fun
Again this seems to be accurate.
Also, I harnessed the breeze, you're welcome
To fill your sails and shake your trees
Māui is indeed depicted as harnessing the wind.
At this point in Moana, Māui's tattoos are showing mini-Maui flying a kite:
This is interesting as there's an entire myth about Māui flying a kite - so that's right, too!
So, what can I say except "you're welcome"?
For the islands I pulled from the sea
Māui did indeed pull up a lot of islands, including New Zealand!
The tide, the grass, the ground
Oh, that was Maui[sic]just messing around
All lies. The tides, as we established earlier, are not Māui's responsibility  . The grass and the ground would appear to be under Papahānaumoku's domain, as she's the earth mother (more on her later!).
I killed an eel, I buried its guts
Sprouted a tree, now you've got coconuts
This seems to be an actual legend. It's made worse, though, as the eel is his wife's lover in the legend.
What is Māui without his hook? (I can't tell you. It's a secret.) But in the movie, he states that it's a gift from the Gods:
Moana: We can fix it!
Māui: It was made by the Gods. You can't fix it. 
This doesn't appear to be matched up with the myths; from what I can find, he made Manaiakalani, the hook, out of the jawbone of his ancestor. Yes, eww.
Throughout the Moana movie, the hook gives Māui the power to transform. He seems to have the ability to transform in mythology as well, he turns into a bird a couple of times .
He's described as a trickster; this seems accurate. He pulled up the islands by tricking his brothers, and he's described as a trickster other places.
Te Ka is interesting - she doesn't really have anything to be compared against. The closest, though, is probably Pele - the goddess of volcanoes.
Te Ka is a lava demon in this form; and female. Pele is also a female goddess, and that's about where the similarity ends. They're both giant female volcano beings.
Interestingly, the Samoan version of Māui, Ti'iti'i, has a fight with Mafui'e, the god of earthquakes and fire, and breaks off one of his arms. At 1:26:.. in Moana, Māui cuts off one of Te Ka's arms (and later cuts off the other one) (YouTube link).
Te Fiti would appear to be directly modeled on Papahānaumoku, the mother island goddess. But the whole thing about her heart seems to have been created by Disney. I can't find a thing about her having a special heart in the mythology. She is certainly the mother island goddess, though, so although the name differs, there's certainly a basis for this in the mythology.
Tamatoa and the Realm of Monsters
I couldn't find a thing about either one of these. It seems Disney may have made these up. There are a bunch of monsters in Polynesian mythology, called 'taniwha', but I was unable to find a reference to them having a world to themselves. I couldn't find anything about giant crabs, though. I did find out that 'Tamatoa' is the name of a Tahiti king .
Tatatoa bears some similarity to the giant crab that Hercules in Greek mythology defeated, though – at least, they’re both giant crabs.
In mythology, the Kakamora are indeed evil - or at least malicious  , according to some (others say they're mostly harmless), but they aren't coconuts - they're just short people. In fact, according to the Carlin Brothers, in the movie, they're not even coconuts at all - they're just wearing coconut armor.
The mythology in Moana (mostly) had backings in the original Polynesian myths, but Disney did invent several things and changed several details.
Maori with enough social standing to tat up had their faces marked to indicate rank and give a visual run-down of one’s accomplishments, position, ancestry, and marital status, among other pieces of socially relevant information.
Maui's birth is generally not dissimilar to that of his other brothers, in New Zealand the hero is declared to have been an abortion, which his mother wrapped up in her apron or topknot, and either abandoned in the bush or threw into the sea. Although thus deserted by his parent, Maui survived, for the unformed child was tended by supernatural beings and reared to manhood, some versions declaring that he was taken up into the sky-world.
 All lyrics from "You're Welcome" are taken from genius.com.
In some versions, Tangaroa has a son, Tinirau, and nine daughters (1891:463). As Tangaroa-whakamau-tai he exercises control over the tides.
: Courtesy of JaneResture.com:
God of the ocean who breathes only twice in 24 hours thus creating the tides.
: Bits of Spirit & Parts of Soul"...Reclaiming the Archetypes of Creation Within, by Nevine Z Rottinger, (available on Google Books):
In Polynesian mythology, Tangaroa is the sea god, father of all sea creatures. The tides are his breaths; one inhalation and one exhalation every twenty four hours.
: From Wikipedia:
Kāne is the creator and gives life associated with dawn, sun and sky. No human sacrifice or laborious ritual was needed in the worship of Kāne.
: Sacred-Texts includes 'Kane of the sun' in their names for him:
 Wikipedia claims, this, actually:
One day, Māui realized that men were being constrained by the sky. The sky was too low and people were not able to stand upright. Māui felt terrible when he saw the people of Earth suffering from this and wanted to help. So Māui searched for his father in order to help him raise the sky so that the men would not suffer from the falling sky.
Māui traveled to the town Lahaina in order to meet his father and push the sky up. Māui then lay parallel to the sky in order to brace himself and push the sky up with his great power. Māui then gave the signal to his father to start pushing the sky up as well, and the strength of father and son together was able to push the sky up high enough for the people of the earth to be able to continue doing daily tasks. Some say if Māui and his father, Ru, had not worked together, the sky would have fallen completely and made the earth uninhabitable for humans. Thus, they saved mankind.
 A collection of myths from around the world, Windows2Universe.org says this:
The trickster god was always trying to impress women. According to one myth, Maui was making an earth oven when his poker got stuck in the sky. At that time, the sky was much lower than it is now. To get more room, Maui simply pushed the sky up. He did this to impress a lady.
Sacred Texts to the rescue again:
The deed is here accomplished in a rather commonplace manner, wholly by Maui, or Tiitii, as he is called in Samoa, and no question of any deity whatever is involved. In Hawaii no other form of the episode seems to exist, but in Samoa there are several variants, according to which the sky is raised by another being at the behest of Tangaloa. Two types appear in the remainder of central Polynesia from which we have material available. There is, first, that where the action is attributed to one of the deities, usually Ru; and secondly, that form which ascribes the deed to Maui, aided by Ru.
On another occasion Maui was out walking and came upon a girl who complained that the sky was so low it kept falling on her and preventing her from doing her chores. Eager to impress the girl, Maui pushed hard and succeeded in raising the sky.
Sacred Texts is quite helpful, isn't it?
The third of the great exploits usually accredited to Maui is that of the fire-quest.
The fire being out, his mother was about to send a servant to secure some, when Maui volunteered to bring it and accordingly went to the house of his ancestor Mafuike, an old woman who was the owner and guardian of fire. Of her he begged a brand, and she gave him one of her fingers, in which fire was concealed. He started away, but when out of sight, quenched it in a stream and returned for more. She gave him another finger, which he extinguished in a similar manner, and thus got from her in succession all her fingers and toes, except the last, with which, in anger, she set the world afire.
It's a different Sacred Texts article:
The usual legend makes Maui the one who takes fire away from Mafuia. The story of fire finding in Polynesia sifts itself to Maui under one of his widely-accepted names, or to his father or to his ancestress-with but very few exceptions.
In some versions, she is the younger sister of Hine-nui-te-pō, goddess of death. It was from her that Māui (in some versions he is her grandson) obtained the secret of making fire.
Sacred Texts yet again...:
A Hawaiian version of the snaring of the sun may be taken as an example of Maui's next exploit.
**yawns** Back at that Sacred Texts site again. Ya know, maybe I'll just use it the whole way through...
The capture and imprisonment of the winds is one of the minor feats often attributed to Maui in New Zealand, where he is said to have caught and confined in caves all but the west wind, which eluded him. In Samoa the winds are gathered up and put in a canoe or coco-nut; while in the Chatham Islands they are collected in a basket, not by Maui, but by another hero, Tawhaki.
 A lot of footnotes here referencing Sacred Texts...
The Hawaiian myths are perhaps the only ones of the Pacific Ocean which give to any of the gods the pleasure and excitement of kite flying. Maui, after repeated experiments, made a large kite for himself.
Okay, I'll be lazy and just source Wikipedia. Sue me.
The great fish-hook of Māui is called Manaiakalani and it is baited with the wing of Hina's pet bird, the ʻalae. Māui is said to have created Hawaii's islands by tricking his brothers. He convinces them to take him out fishing, but catches his hook upon the ocean floor. He tells his brothers that he has caught a big fish, and tells them to paddle as hard as they can. His brothers paddle with all their might, and being intent with their effort, did not notice the island rising behind them. Māui repeats this trick several times, creating the Hawaiian Islands
There's a NewZealand.com? TIL.
The hook went deeper and deeper into the sea until Māui felt the hook had touched something. He tugged gently and far below the hook caught fast. It was a huge fish! Together with his brothers, Māui brought the fish to the surface.
Māui cautioned his brothers to wait until he had appeased Tangaroa the god of the sea before they cut into the fish. They grew tired of waiting and began to carve out pieces for themselves. These are now the many valleys, mountains, lakes and rocky coastlines of the North Island.
From a paper on the Kamehameha Schools site:
According to Hawaiian tradition, the islands were born (for the
most part) from the union of Wäkea, often referred to in other cultures as
Sky Father, and Papahänaumoku, the earth mother.
From Sacred Texts:
The other episode is that where Maui kills Tuna, the eel lover of his wife. The latter went one day to the stream to get water, and while she stood on the bank, Tuna came up in the guise of a great eel, struck her with his tail, knocked her into the stream, and maltreated her. Angry at this, Maui laid down two logs on which Tuna might cross over, and then, hiding, killed the eel as he came, after which various plants, trees, fish, and monsters of the deep were derived from the creature's head and body.
Not the best transcript out there, but from moana.wikia.com, with some slight edits from me.
Maui offered to take their place, but when he came to his ancestress, he found her ill, one half of her body being already dead, whereupon he wrenched off her lower jaw, made from it a fish-hook, which he concealed about him, and then returned to his home.
But his brothers said, 'Man cannot go near to the sun on account of the heat.' Maui said, 'You have seen the many acts that I have performed. I have taken the form of a bird, and again resumed that of a man, while you have ever had the form of men. And now, my brothers, I can do what I propose, and even greater acts than this.'
A different Sacred Texts:
Maui planned to follow her, but first studied the forms of birds that he might assume the body of the strongest and most enduring. After a time he took the shape of a pigeon and, flying to the black rock, passed through the door and flew down the long dark passage-way.
Māui is said to have created Hawaii's islands by tricking his brothers.
MythEncyclopedia, for instance:
In Polynesian mythology, Maui was a powerful trickster god best known for creating the Pacific islands.
Yeah, just going off of Wikipedia here, sorry 'bout that.
In Samoan legend, the mythological figure Ti'iti'i appears in legends very similar to those recounting the tales of the demigod Māui, found in other island cultures. In one such legend, which is almost identical to the New Zealand fire myth of Māui, he succeeds in bringing fire to the people of Samoa after a battle with the earthquake god, Mafui'e.
Sacred Texts tells us this:
What a battle there was for a time in the underworld! At last Ti'iti'i seized one of the arms of Mafuie and broke it off. He caught the other arm and began to twist and bend it.
Mafuie begged the boy to spare him. His right arm was gone. How could he govern the earthquakes if his left arm were torn off also? It was his duty to hold Samoa level and not permit too many earthquakes. It would be hard to do that even with one arm-but it would be impossible if both arms were gone.
Papahānaumoku, sometimes called Papa, is the earth mother goddess in Hawaiian religion of the Kanaka Maoli. Together with her husband Wākea (sky father) Papa is the ancestor of all people and Kalo, and mother of islands as the Kanaka Maoli manifestation of Mother Earth.
In Māori mythology, taniwha (Māori pronunciation: [ˈtanifa]) are beings that live in deep pools in rivers, dark caves, or in the sea, especially in places with dangerous currents or deceptive breakers (giant waves). They may be considered highly respected kaitiaki (protective guardians) of people and places, or in some traditions as dangerous, predatory beings, which for example would kidnap women to have as wives.
Not really related to the question, but... from Wikipedia:
Tamatoa V, born Tamatoa-a-tu Pōmare, (23 September 1842, Moorea – 30 September 1881, Pape'ete), King of Raiatea and Taha'a, was a son of Queen Pōmare IV of Tahiti.
Seeing that Hercules was winning the struggle, Hera sent a giant crab to distract him. He crushed it under his mighty foot.
From Sacred Texts again:
Two classes of spirits are described on San Cristoval, distinct but sometimes confused with each other. The Kakamora are said to be from six inches to three or four feet in height, from fair to dark, go naked with long straight hair to their knees, are strong as three or four men, and fond of dancing and singing. They do not use cooked food. They have a ruler, male or female. They are described as harmless but tricky, or as malicious and dangerous, and are differently named all along the coast.
From a site called GodChecker.com:
These are legendary Oceanic creatures a couple of feet high with long sharp fingernails. They have been described as hobbit-like [..]
The KAKAMORA tend to hide in caves and prey on stray children and travelers. They can be kept at bay by waving something pale. Far from being a token of surrender, white absolutely terrifies them.
On a site devoted to the Solomon Islands, they say this:
According to descriptions of two British anthropologists who did research in Solomon Islands in 1920s, Dr C.E. Fox and Dr F.H. Drew, "Kakamora [...] are not quite human. They vary in height, from six inches [approx. 15 cm - WB] to three or four feet [approx. 95 to 125 cm - WB] [...]. Most of them are considered to be quite harmless, but sometimes they have been known to attack men. When they do so they use their fingers, which are furnished with long sharp nails with which they stab. [...]"
You can watch the video here. As proof, they include concept art.