Let's begin this answer with a quote from GRRM:
The short answer is that the laws of inheritance in the Seven Kingdoms
are modelled on those in real medieval history... which is to say,
they were vague, uncodified, subject to varying interpretations, and
often contradictory.
With that being said, Cersei is the current Lady paramount of the Westerlands by both the show and the books.
Kevan and his proginy can't ascend to throne of the Kings of the Rock while a Child of Tywin's body is alive and free to claim his/her rights.
Lannisters are Andals who practice faith of seven. According to Andal traditions, rights of a son come before an elder daughter. Similarly rights of a daughter come before rights of an uncle. So the claimants with their number in Line of succession to Throne of Kings of the Rock would be (Ideally):
╔════════════════════╦════════════════╦══════════════════╗
║ Claimaint ║ Number in Line ║ Remarks ║
╠════════════════════╬════════════════╬══════════════════╣
║ Jaime Lannister ║ 1 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Tyrion Lannister ║ 2 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Cersei Lannister ║ 3 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Joffrey Baratheon ║ 4 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Tommen Baratheon ║ 5 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Myrcella Baratheon ║ 6 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Kevan Lannister ║ 7 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Lancel Lannister ║ 8 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Martyn Lannister ║ 9 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Willem Lannister ║ 10 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Janei Lannister ║ 11 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Tygett Lannister ║ 12 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Tyrek Lannister ║ 13 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Gerion Lannister ║ 14 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Joy Hill ║ 15 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Genna Lannister ║ 16 ║ Ideal Conditions ║
╠════════════════════╬════════════════╬══════════════════╣
║ Cleos Frey ║ 17 ║ Ideal Conditions ║
╚════════════════════╩════════════════╩══════════════════╝
But in reality and by legal impacts, the line of succession becomes this as per show:
╔════════════════════╦════════════════════════╦══════════════════════╦══════════════════════════════════════════════════╗
║ Claimaint ║ Number in Line (Books) ║ Number in Line(Show) ║ Remarks ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Jaime Lannister ║ NA ║ NA ║ Disinherited as per vows of KG- See spoiler ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Tyrion Lannister ║ NA ║ NA ║ Disinherited and wanted for murder of his father ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Cersei Lannister ║ Incumbent ║ Incumbent ║ Current Lady ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Joffrey Baratheon ║ NA ║ NA ║ Dead ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Tommen Baratheon ║ 1 ║ 1 ║ See Spoiler at bottom ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Myrcella Baratheon ║ 2 ║ NA ║ Dead as per show, alive in books ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Kevan Lannister ║ 3 ║ 2 ║ See Spoiler at bottom ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Lancel Lannister ║ NA ║ NA ║ Disinherited due to joining the Sparrows ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Martyn Lannister ║ 4 ║ NA ║ Killed by Lord Karstark in show, alive in books ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Willem Lannister ║ NA ║ NA ║ Killed by Lord Karstark ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Janei Lannister ║ 5 ║ NA ║ Not mentioned in Show ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Tygett Lannister ║ NA ║ NA ║ Dead ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Tyrek Lannister ║ NA ║ NA ║ Missing, presumed Dead. Not mentioned in show ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Gerion Lannister ║ NA ║ NA ║ Dead ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Joy Hill ║ NA ║ NA ║ Bastard - Ineligible to inherit ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Genna Lannister ║ 6 ║ 3 ║ Tywin's sister, don't know if mentioned in show ║
╠════════════════════╬════════════════════════╬══════════════════════╬══════════════════════════════════════════════════╣
║ Cleos Frey ║ NA ║ NA ║ Dead ║
╚════════════════════╩════════════════════════╩══════════════════════╩══════════════════════════════════════════════════╝
By filtering out dead and disinherited characters, we get the following line:
╔════════════════════╦════════════════════════╦══════════════════════╦════════════════════════════╗
║ Claimaint ║ Number in Line (Books) ║ Number in Line(Show) ║ Remarks ║
╠════════════════════╬════════════════════════╬══════════════════════╬════════════════════════════╣
║ Cersei Lannister ║ Incumbent ║ Incumbent ║ Current Lady ║
╠════════════════════╬════════════════════════╬══════════════════════╬════════════════════════════╣
║ Tommen Baratheon ║ 1 ║ 1 ║ See spoiler at bottom ║
╠════════════════════╬════════════════════════╬══════════════════════╬════════════════════════════╣
║ Myrcella Baratheon ║ 2 ║ NA ║ Dead according to show ║
╠════════════════════╬════════════════════════╬══════════════════════╬════════════════════════════╣
║ Kevan Lannister ║ 3 ║ 2 ║ See spoiler at the bottom ║
╠════════════════════╬════════════════════════╬══════════════════════╬════════════════════════════╣
║ Martyn Lannister ║ 4 ║ NA ║ Dead according to the show ║
╠════════════════════╬════════════════════════╬══════════════════════╬════════════════════════════╣
║ Janei Lannister ║ 5 ║ NA ║ Not mentioned in the show ║
╠════════════════════╬════════════════════════╬══════════════════════╬════════════════════════════╣
║ Genna Lannister ║ 6 ║ 3 ║ NA ║
╚════════════════════╩════════════════════════╩══════════════════════╩════════════════════════════╝
Jaime can't be lord because of his vows. Tyrion is an attainted traitor and wanted murderer besides. That leaves us with Cersei. With her death, the seat will pass to Tommen No-Kidding-Baratheon unless Tyrion lands in Westeros with Daenerys and makes sure that his sister does die.
In the books, she appoints one of her cousins as acting Warden of the West and another cousin as Castellan of Casterly Rock while she resumes regency of Iron Throne in KL. She refuses to go to Casterly Rock when her uncle reminds her that she is the lady of the Rock and her place is there.
Can't say if they have included this in show but even in show wiki, Cersei is the current lady of Casterly Rock.
Following are two family trees providing a visual info of claimants and their status.
Line of Succession as per Show
Line of Succession as per Books
Spoiler Alert:
According to Epilogue of A Dance with Dragons, Kevan is dead at
hands of Varys. Which means he is also out of the race with his son
Martyn now 3rd in LoS and daughter Janei 4th. His sister Genna ascends to 5th
Show Spoilers for Season 6 Episodes 6 & 10:
Tommen dismissed Jaime from Service as Kingsguard. He is now eligible
to inherit the Rock and thus is the heir apparent as per show canon given that Tommen has committed suicide and is thus eliminated from the line. Lancel was also killed in destruction of Sept of Baelor.
Show Spoilers for Season 8 Episodes 6:
Cersei and Jaime died together along with their unborn baby. Which makes Lord Hand, Tyrion Lannister the last Lannister. Since he was cleared of all crimes and granted the position of the Hand under King Bran, it is evident that he is no longer attainted and is therefore the Rightful Lord of Casterly Rock and Lord Paramount of Westerlands, especially since we have no information otherwise.
EDIT: I did some further research w.r.t Books in order to find out Lannister claimants other than line of Tytos Lannister and link the other known Lannisters to the line.
Bear in mind that answer above deals only with the Line of Tytos Lannister.
Now moving back in Lannister family tree, We see that Grandfather of Cersei, Jaime and Tyrion, Lord Tytos Lannister had another brother named Ser Jason Lannister.
Ser Jason fathered numerous children which include:
- Joanna Lannister who married her cousin, Tywin Lannister. Tywin was at that time heir to the Casterly Rock and Hand of the King. She died giving birth to Tyrion.
- Ser Stafford Lannister. He was killed by Lord Rickard Karstark in Battle of Oxcross.
- Damon Lannister. It is unknown whether he is alive or dead.
- Lynora Hill. She is presumably alive however it is unknown who she married or if she had children.
- 2 unnamed sons and 2 unnamed daughters.
Joanna, Stafford and Damon all had children and two of them at least had grandchildren. Joanna's children are of course the senior Branch and main line (Jaime, Cersei, Tyrion) along with their own descendants (Tommen and Myrcella).
Children of Stafford and Damon are listed below:
- Stafford Lannister
- Daven Lannister
- Cyrenna Lannister
- Myriella Lannister
- Damon Lannister
- Damion Lannister
- Lucion Lannister
- Lanna Lannister
Lanna Lannister is married to Lord Jast and her children are unknown along with children of her cousins.
So according to books, after the line of Genna Lannister and her Frey descendants, we get the following (known) people in line of succession:
╔════════════════════╦════════════════════════╦══════════════════════╦═════════════════════════════════╗
║ Claimaint ║ Number in Line (Books) ║ Number in Line(Show) ║ Remarks ║
╠════════════════════╬════════════════════════╬══════════════════════╬═════════════════════════════════╣
║ Damon Lannister ║ 7 ║ NA ║ Eldest son of Jason Lannister ║
╠════════════════════╬════════════════════════╬══════════════════════╬═════════════════════════════════╣
║ Damion Lannister ║ 8 ║ NA ║ NA ║
╠════════════════════╬════════════════════════╬══════════════════════╬═════════════════════════════════╣
║ Lucion Lannister ║ 9 ║ NA ║ NA ║
╠════════════════════╬════════════════════════╬══════════════════════╬═════════════════════════════════╣
║ Lanna Lannister ║ 10 ║ NA ║ Married to Lord Jast ║
╠════════════════════╬════════════════════════╬══════════════════════╬═════════════════════════════════╣
║ Daven Lannister ║ 11 ║ NA ║ NA ║
╠════════════════════╬════════════════════════╬══════════════════════╬═════════════════════════════════╣
║ Cyrenna Lannister ║ 12 ║ NA ║ NA ║
╠════════════════════╬════════════════════════╬══════════════════════╬═════════════════════════════════╣
║ Myriella Lannister ║ 13 ║ NA ║ NA ║
╠════════════════════╬════════════════════════╬══════════════════════╬═════════════════════════════════╣
║ Lynorra Hill ║ NA ║ NA ║ Bastard - ineligible to inherit ║
╚════════════════════╩════════════════════════╩══════════════════════╩═════════════════════════════════╝
A visual description of Jason's line will be:
Bear in mind, none of these characters are mentioned in the show.