Cersei can't establish her claim on Iron Throne. She is the Dowager Queen and Queen Mother, nothing more than that. She has claimed the Iron Throne because she could.
King Aegon V Targaryen's daughter Princess Rhaelle married Lord Ormund Baratheon. They had a son named Steffon who is father of Robert, Renly and Stannis. Robert's claim to Iron Throne came due to his descent from King Aegon V.
Cersei has no such claim as she is not descended from Targaryens.
Grand Council to elect a new King
Legally, in event of ambiguous circumstances in succession and complex case of finding the heir apparent, A Grand Council of all the lords of the realm is called where lords choose the King or heir by popular vote. All claimants can have their say in the council.
Great Council of 101 AC
King Jaehaerys's eldest sons Princes Aemon and Baelon predeceased their father.
Since His Grace's only surviving Son Vaegon was a maester, he could not choose Vaegon as his successor because maesters are forbidden to hold lands and titles by vows. This meant he would have to choose a successor from his grandchildren; Princess Rhaenys (Only daughter of the eldest child of King Jaehaerys, Prince Aemon), or Prince Viserys, Prince Daemon (Sons of the the King's second son Baelon). Jaehaerys may have wanted to choose Viserys as Daemon was younger than Viserys and Rhaenys was a woman but he knew that explicit royal edict in this regard would result in civil war on his death as Rhaenys' maternal uncle Lord Boremund Baratheon and her husband Lord Corlys Velaryon were two of the strongest Lords in the seven Kingdoms and would never have stood aside meekly for Prince Viserys to take the throne when it belonged to Rhaenys by rights.
So to avoid prospects of another civil war (As Jaehaerys himself took the throne after a civil war with his uncle King Maegor the Cruel) by getting a consensus of Lords of realm, Jaehaerys called a grand council to choose between his son Aemon's daughter Rhaenys1, Baelon's son Viserys and Rhaenys' son Laenor Velaryon.
Tensions ran high which proved that Jaehaerys' concerns weren't unfounded. Lord Baratheon and Lord Velaryon actively lobbied in support of Rhaenys' son Laenor. Prince Daemon started gathering soldiers to take the throne forcibly for his elder brother Viserys if council chose Laenor or Rhaenys.
As a result, Rhaenys was passed over because she was a woman and never made her claim , Laenor was passed over because he was a child despite support of Lord Boremund Baratheon and Lord Corlys Velaryon, thus Prince Viserys was chosen as heir of his grandfather King Jaehaerys I.
Great Council of 233 AC
King Maekar I's eldest son Daeron predeceased his father. Second son Prince Aerion also predeceased King Maekar. Daeron left a young girl and Aerion left a baby boy.
Thus on unexpected death of King Maekar, the King's hand Lord Brynden Blooraven called a great council. Claims of Daeron's daughter2, Aerion's son3 & Aenys Blackfyre4 were dismissed. Lords of the realm reluctantly elected youngest son of King Maekar, Prince Aegon5.
Thus it was Prince Aegon who took the throne as King Aegon V.
In show canon, House Baratheon is now legally extinct because
Tommen was the last (Apparent) Baratheon after deaths of Stannis
Baratheon, Shireen Baratheon and Myrcella Baratheon. Renly Baratheon
had already died.
This means only unacknowledged bastard Baratheons are alive e.g. Gendry. There are no more Baratheons.
Also Robert Baratheon's great grandfather, Lord Lyonel Baratheon fathered at least one daughter and son Ormund Baratheon. Ormund Baratheon sired Steffon Baratheon who is supposedly only child of marriage between Ormund Baratheon and Rhaelle Targaryen. Steffon's line ended with Tommen (who is in fact not part of the line anyway). So we have to go back to that daughter of Lord Lyonel Baratheon. We do not know her name or which house she married in. But in event of a grand council, her descendents would have the best claim as heirs to House Baratheon.
Not to mention, there might be cadet dynasties of House Baratheon/Durrandon such as House Bolling and House Wensington who will be claimants as well in case of extinction of House Baratheon. Also, in the books, there is Edric Storm as well, the only acknowledged bastard of Robert Baratheon. If the Lords of the realm agree to it (Assuming such a situation rises in the books as well), he can be legitimized by a council or simply crowned and left to legitimize himself.
Jaime is not Lord of the Casterly Rock, Cersei is the lady of the Casterly Rock.
Jaime is sworn brother of Kingsguard and thus is disqualified from inheriting any lands. Jaime was dismissed by King Tommen in S06E06 and is thus qualified for inheritence again, as pointed out by BCdotWeb.
Tommen: When you attack the Faith, you attack the Crown. Anyone who
attacks the Crown is unfit to serve as Lord Commander of the
If Jaime is not a KG anymore, that makes him Heir to Casterly Rock, not Lord of Casterly Rock. Jaime was still a KG when Tywin Lannister died, which is why Cersei inherited the Lannister lands. Cersei may however choose to abdicate as Lady of Casterly Rock and give it to Jaime.
The Only surviving legal direct claimant to the Throne
Unless a Great council is convened and elects a new candidate, the only legal claimant to Iron Throne of Aegon I is now Daenerys Stormborn.
Jon Snow has been revealed to be a Targaryen but
it is unclear whether
he is trueborn or a bastard. It has been revealed in Season 7 that Jon is not a bastard but rather trueborn son of Rhaegar and Lyanna (Named Aegon at birth), making him the real heir to the Throne, ahead of his aunt Daenerys in line of succession.
Daenerys has sailed from Meereen to take back her father's throne. From her POV, Robert was a usurper and so were his successors. Many in the Seven Kingdoms would agree with her. So even if a council is convened, Daenerys has the strongest claim being daughter of King Aerys II.
That is unless Jon turns out to be trueborn son of Rhaegar Targaryen.
If that happens, Jon will have the strongest claim. Jon is trueborn son of Rhaegar But Daenerys has
dragons so I suppose it does not matter.
What can we speculate?
This means succession is not clear and a grand council is in order. Cersei however took the throne for her own but that does not mean that other lords would accept her claim meekly.
Most likely this means uprising of the lords who had previously acknowledged authority of King Tommen Baratheon and House Baratheon. Reach and Dorne will be the chief Kingdoms to rise up because they have the means and power to defy Cersei. They have already added their power to Daenerys'.
Seeing how High septon and seat of the faith were destroyed by Cersei, it is unlikely that new High Septon who will be most likely elected in Starry Sept, Old Town, Reach6 now (And thus under Tyrell influence), will support or tolerate Cersei. She will be branded a heretic and a criminal against gods.
Both factions of House Greyjoy are already seeking to ally themselves to Daenerys Targaryen. Yara Greyjoy has succeeded in winning Daenerys over to her side so Euron either has to fight alone and die or he can join the Lannisters and form marital alliance with Cersei instead.
Riverlands are in disarray and Northmen are busy in their own succession squabbles and in solidifying the reestablished Stark dominion.
And the greatest threat of all, Daenerys Stormborn, now the only claimant to the throne other than Jon "Snow" Targaryen is already on her way to Westeros. She can directly go to Old Town and Starry Sept (Without any fighting as Tyrells are committed to her cause) and have the new High Septon crown and anoint her as Queen Daenerys of House Targaryen, first of her name, Queen of the Andals and the Rhoynar and the first Men just like her ancestor Aegon the Conqueror was crowned there.
This means the war that started after Robert's death is far from over.
"Vengeance, Justice, Fire and Blood" is coming.
1 Jaehaerys had actually passed over his grand-daughter Rhaenys in favor of his second son Baelon when her father the Prince of Dragonstone Aemon died. So in GC of 101 AC, Rhaenys wasn't actually a choice nominated by Jaehaerys I for the GC. She was however a claimant and thus reserved the right to make her case in the Grand council. She however, chose not to, and instead supported her son Laenor Velaryon's claim which came from her. She was nicknamed "Queen who never was" by smallfolk because technically she should have succeeded her grandfather as Queen of Westeros.
2 Princess Vaella Targaryen, daughter of Prince Daeron Targaryen, was simple-minded. On account of her mental incapacity and her gender, her claim was dismissed after brief consideration.
3 Prince Maegor Targaryen, son of Prince Aerion Targaryen, was an infant. Only a few lords supported his claim but most spoke against it because they feared that the infant may have inherited his father's madness. Many also opposed his claim because he was named Maegor, after the cruelest Targaryen King.
4 Aenys Blackfyre was actually executed by Brynden 'Bloodraven' Rivers before he could make his claim. This was a breach of promises made to Aenys by Lord Bloodraven which guaranteed Aenys' safe passage. Young Prince Aegon Targaryen was especially infuriated by this treachery. So when Council chose Prince Aegon as King Aegon V, his first act was to condemn Bloodraven to death. Later the King relented and offered Bloodraven a chance to take the Black. Bloodraven accepted and went on to become Lord Commander of Night's Watch eventually. He retained that position and last Targaryen ancestral blade Darksister until his disappearance beyond the Wall.
5 Lords first quietly offered the crown to Maester Aemon who had taken vows of a Maester and forsaken his name and claims. Since many lords were against his younger brother Prince Aegon, they decided to overpass Aegon's claim and offer the throne to Aemon instead. High Septon supported those lords and offered to absolve Aemon of all vows if he would agree to take the crown. Aemon however refused to dishonor his vows and took the black when his younger brother Aegon was crowned.
6 Before King Baelor the Blessed built Great Sept of King's Landing (Later named after the King), Starry Sept used to be seat of the Faith. Aegon the Conqueror was crowned there by High Septon when he defeated allied Armies of Westerlands, The Reach and the North. Since Great Sept of Baelor is now destroyed as per show canon, the Faith council will now probably meet in second most prestigious sept in Westeros, which is Starry Sept, to choose the new High Septon.