This question is based on The Lord of the Rings books. In the books, Aragorn planned to go to Gondor and aid them in their fight against Mordor. Aragorn knew Denethor from his time at Gondor serving Ecthelion II. He might have known that he might not be welcome there.
From Appendix A and the "Tale of Years", we know that Aragorn served Ecthelion II in disguise. This time, he intended to reveal his true identity, so we can be fairly sure he planned to become king.
His [Aragorn's] own plan, while Gandalf remained with them, had been to go with Boromir, and with his sword help to deliver Gondor. For he believed that the message of the dreams was a summons, and that the hour had come at last when the heir of Elendil should come forth and strive with Sauron for the mastery.
("Farewell to Lórien")
Boromir accepted his help because he knew that Gondor was losing the war.
Yet we are hard pressed, and the Sword of Elendil would be a help beyond our hope — if such a thing could indeed return out of the shadows of the past.
("The Council of Elrond")
The Heirs of Isildur had long memories, never forgetting that they had once ruled much of Eriador as the heirs of Elendil. In a world in which prophesies are known to come true, the words of Malbeth the Seer were remembered:
"Arvedui you shall call him, for he will be the last in Arthedain. Though a choice will come to the Dúnedain, and if they take the one that seems less hopeful, then your son will change his name and become king of a great realm. If not, then much sorrow and many lives of men shall pass, until the Dúnedain arise and are united again."
--- "Gondor and the heirs of Anárion" (The Lord of the Rings, Appendix A, Part IV)
Aragorn, although he tells Boromir that he looks little like the statues of Isildur and Anárion in Minas Tirith, is mentioned as looking almost identical to Elendur — who is described as looking almost the same as his grandfather, Elendil. No doubt this was taken as a sign of great things ahead for Aragorn II.
So perished Elendur, who should afterwards have been King, and as all foretold who knew him, in his strength and wisdom, and his majesty without pride, one of the greatest, the fairest of the seed of Elendil, most like to his grandsire.26
Footnote 26: It is said in later days those (such as Elrond) whose memories recalled [Elendur] were struck by the great likeness to him in body and mind, of King Elessar, the victor in the War of the Ring
--- "The Disaster of the Gladden Fields" (Unfinished Tales)
In the century leading up to the War of the Ring, it was known by those on both sides that things were coming to a head. Sauron had returned to Mordor. Elves were heading to the havens, Orcs were multiplying. Gondor's strength was waning, and it was losing territory to Mordor and Harad. Meanwhile, Sauron was searching to find any trace of the man who, if he existed, would unite his enemies against him.
Aragorn, being now the Heir of Isildur, was taken with his mother to dwell in the house of Elrond; and Elrond took the place of his father and came to love him as a son of his own. But he was called Estel, that is "Hope", and his true name and lineage were kept secret at the bidding of Elrond; for the Wise then knew that the Enemy was seeking to discover the Heir of Isildur, if any remained upon earth.
--- "The Tale of Aragorn and Arwen" (The Lord of the Rings, Appendix A, Part V)
When Elrond revealed to Aragorn his true name and ancestry and gave to him the heirlooms of his house, he foresaw that Aragorn would be tested, and his labours would be great. Since the fall of Arthedain, Elrond had been custodian of the Crown Jewels of Arnor, most notably, the Sceptre of Annúminas — and he specifically said that Aragorn was yet to earn it, a phrasing I think it unlikely he would have used for the 14 Chieftains of the Dunédain fostered in Rivendell before Aragorn.
"Here is the ring of Barahir," he said, "the token of our kinship from afar; and here also are the shards of Narsil. With these you may yet do great deeds; for I foretell that the span of your life shall be greater than the measure of Men, unless evil befalls you or you fail at the test. But the test will be hard and long. The Sceptre of Annúminas I withhold, for you have yet to earn it."
--- "The Tale of Aragorn and Arwen" (The Lord of the Rings, Appendix A, Part V)
When Aragorn was 49 years old, he met Arwen, Elrond's daughter, for the second time. I'm not 100% on the old language that Tolkien uses in this passage, but best I can tell, they got engaged (and also Arwen agreed to lay aside her immortality for him.) Elrond found this out, and he was displeased.
When Aragorn next came to Rivendell, Elrond tells him says that unless he restores the Númenórean Realms-in-Exile, Aragorn won't be marrying his daughter.
Maybe, it has been appointed so, that by my loss the kingship of Men may be restored. Therefore, though I love you, I say to you: Arwen Undómiel shall not diminish her life's grace for less cause. She shall not be the bride of any Man less than the King of both Gondor and Arnor.
--- "The Tale of Aragorn and Arwen" (The Lord of the Rings, Appendix A, Part V)
That encounter happened about 35 years before Aragorn meets Frodo at the Prancing Pony, and it's directly after his time in the service of Rohan and Gondor in the guise of Thorongil. During those years, he works with Gandalf to find Gollum, protect the Shire, and who knows what else, in order to overthrow Sauron.
And then we come to the Council of Elrond, where Aragorn asks Boromir:
Do you wish for the House of Elendil to return to the Land of Gondor?
Coming openly to Gondor, declaring his name and position, even only as Chieftain of the Dunédain of Arnor, can only mean one thing: Claiming the Crown, and taking Arwen's hand in Marriage.
And that brings me to the second part of your question:
Boromir came to Rivendell after he and and brother Faramir receive a vision. A strange man stands up and claims he carries the Shards of Narsil, then asks if Boromir would have the House of Elendil return to Gondor after almost 1,000 years of Ruling Stewards.
"I was not sent to beg any boon, but to seek only the meaning of a riddle" said Boromir proudly. "Yet we are hard pressed, and the Sword of Elendil would be a help beyond our hope — if such a thing could indeed return out of the shadows of the past." He looked again at Aragorn, and doubt was in his eyes.
--- "The Council of Elrond" (The Lord of the Rings, The Fellowship of the Rings)
Bilbo and Aragorn then respond to Boromir's doubts. Aragorn finishes his account of his ancestry by announcing to the Council, without any invitation from Boromir:
But now the world is changing once again. A new hour comes. Isildur's Bane is found. Battle is at hand. The Sword shall be reforged. I will come to Minas Tirith
--- "The Council of Elrond" (The Lord of the Rings, The Fellowship of the Rings)
However Boromir, a forty year old man who has been raised since birth expecting to inherit his father's role as Ruling Steward of Gondor — since the idea of a King finally coming to claim the Throne after a thousand years is plainly ludicrous, remains suspicious. He trusts what he can see, and Aragorn does not impress him.
In Gondor, we must trust to such weapons as we have. And at the least, while the wise ones guard this Ring, we will fight on. Mayhap the Sword-that-was-Broken may still stem the tide — if the hand that wields it has inherited not a heirloom only, but the sinews of the Kings of Men.
--- "The Council of Elrond" (The Lord of the Rings, The Fellowship of the Rings)
Compare this to the reception he receives from Imrahil of Dol Amroth later in the story. The reaction of Faramir and the people of Gondor is similar, (although by this stage Aragorn certainly looks more the part of King than he would have to Boromir at the Council of Elrond.)
As for me ... The Lord Aragorn I hold to be my liege-lord, whether he claim it or no. His wish is to me a command.
--- "The Last Debate" (The Lord of the Rings, The Return of the King)
Even at the end, Boromir does not accept that Aragorn is necessarily the returning King of Gondor, but he has seen that Aragorn is a mighty warrior, and believes that he will do great things against Gondor's enemies. With his dying words he asks Aragorn to save the people of Gondor.
Farewell, Aragorn! Go to Minas Tirith and save my people! I have failed.
--- "The Departure of Boromir" (The Lord of the Rings, The Two Towers)
But at no stage does Boromir use the formality that Imrahil does.
I agree with the existing answers that Aragorn planned to become King of Gondor. He believed it to be his destiny, and it was one of the requirements for Elrond to allow Arwen to marry him
However I don't think it was his plan to immediately challenge Denethor. His immediate goal was to help to defend Gondor. It seems likely that when he reached Gondor, his plan was to do what he did indeed do: fight to defend it and avoid challenging Denethor for as long as Gondor was in danger.
After the Battle of the Pelennor Fields. Aragorn, Éomer and Imrahil have this conversation:
Now as the sun went down Aragorn and Éomer and Imrahil drew near the City with their captains and knights; and when they came before the Gate Aragorn said:
‘Behold the Sun setting in a great fire! It is a sign of the end and fall of many things, and a change in the tides of the world. But this City and realm has rested in the charge of the Stewards for many long years, and I fear that if I enter it unbidden, then doubt and debate may arise, which should not be while this war is fought. I will not enter in, nor make any claim, until it be seen whether we or Mordor shall prevail. Men shall pitch my tents upon the field, and here I will await the welcome of the Lord of the City.’
But Éomer said: ‘Already you have raised the banner of the Kings and displayed the tokens of Elendil’s House. Will you suffer these to be challenged?’
‘No,’ said Aragorn. ‘But I deem the time unripe; and I have no mind for strife except with our Enemy and his servants.’
And the Prince Imrahil said: ‘Your words, lord, are wise, if one who is a kinsman of the Lord Denethor may counsel you in this matter. He is strong-willed and proud, but old; and his mood has been strange since his son was stricken down. Yet I would not have you remain like a beggar at the door.’
‘Not a beggar,’ said Aragorn. ‘Say a captain of the Rangers, who are unused to cities and houses of stone.’ And he commanded that his banner should be furled; and he did off the Star of the North Kingdom and gave it to the keeping of the sons of Elrond.
The Lord of the Rings Book Five, Chapter 7: The Pyre of Denethor
Page 860 (Single volume 50th Anniversary Edition)
Aragorn's answer to Éomer make's clear that he will not allow his claim to be challenged, but he prefers not to press it while there is strife "with our Enemy and his servants".
It seems that Aragorn is being overly optimistic in thinking that Denethor will also be willing to set aside the claim until the war is over. I put that down to the fact that it is many years since he last met Denethor, and Denethor's thinking has been poisoned in recent years by Sauron's influence through the Palantír.
The question also asks why Boromir would accept Aragorn's help. Boromir is more like his father than Faramir, and might have opposed Aragorn's claim. Faramir refers to this when he talks to Frodo at Henneth Annûn:
‘And this I remember of Boromir as a boy, when we together learned the tale of our sires and the history of our city, that always it displeased him that his father was not king. “How many hundreds of years needs it to make a steward a king, if the king returns not?” he asked. “Few years, maybe, in other places of less royalty,” my father answered. “In Gondor ten thousand years would not suffice.” Alas! poor Boromir. Does that not tell you something of him?’
‘It does,’ said Frodo. ‘Yet always he treated Aragorn with honour.’
‘I doubt it not,’ said Faramir. ‘If he were satisfied of Aragorn’s claim, as you say, he would greatly reverence him. But the pinch had not yet come. They had not yet reached Minas Tirith or become rivals in her wars.
The Lord of the Rings Book Four, Chapter 5: The Window on the West
Page 670 (Single volume 50th Anniversary Edition)
I suspect that Boromir's attitude was the same pragmatic one that Aragorn mistakenly expected to find in Denethor: that the war must be fought first, and the rivalry could wait.
