BACKGROUND:
Jaime Lannister @ the Game of Thrones wiki
PERSONALITY:
Jaime Lannister, everybody's favorite Kingslayer, is arrogant, smug, and sarcastic, and all of these things are actually true of him, except when they aren't.
Raised by Lord Tywin Lannister, Jaime and Cersei were both taught the virtues of ruthlessness and familial pride. Amorality was and is encouraged, and doing whatever you have to do to achieve your end is not something to shy from--so long as you are acting in a way that will further bring glory to the Lannister name, or in a way that will protect the family. Cersei is actually better at living for this particular virtue. Jaime isn't bad at amorality, but he has a certain code and a set of personal principles outside of the Lannister name. Largely this has come of being a knight and a member of the Kingsguard--something Jaime once took great pride in. A knight operates by a particular code that Jaime strove to match and be worthy of, and vestiges or versions of that code remain.
The trouble is, Jaime was made a member of the Kingsguard during the reign of King Aerys Targaryen, better known as the Mad King. Jaime was knighted, then kept half a hostage, in order to ensure Lord Tywin's cooperation and neutrality during the first stages of Robert's Rebellion. Kept close to King Aerys, Jaime witnessed firsthand the atrocities that Aerys visited on his own people--members of his Small Council, lords--anyone who dared to question or disobey his crazy whims. And Jaime witnessed just how silent his sworn brothers were willing to be in the face of these same atrocities, all in the name of honor and fealty. All of this has only helped to foster Jaime's deep disillusionment with the dubious virtues of honor and nobility.
The Mad King's reign has shaped Jaime's understanding of violence and its uses. Having witnessed firsthand many and various tortures, Jaime does not tolerate or respect needless violence. He is firm in dealing with those who would act cruel for the sake of cruelness. Violence for a cause, however, is different. This allows Jaime to operate ruthlessly on the battlefield, and ruthlessly push Bran Stark from the window. The latter was not an act that he relished, but a necessary measure to save Cersei and their children from execution. "The things I do for love" isn't just a weary lover's line, it's also an acknowledgement of the shit that Jaime is willing to do to protect Cersei, a verbalization of his own ruthlessness.
In this same vein, at the end of Robert's Rebellion, Jaime took it upon himself to kill the Mad King, which is how he earned his nickname. A member of the Kingsguard is intended to guard the king, so killing the king was completely contrary to that task, and obviously opened Jaime up to a great deal of scorn and disgust from all range of people--fellow nobles, knights, smallfolk. Breaking his sworn word was not the Good thing to do, and Jaime knows it. All the scorn that he has met with has also served to deepen his disillusionment. He broke his word to save all of King's Landing, killed a psychotic king for the betterment of the Seven Kingdoms--and now gets shat on for it. Honor and nobility are only facades.
Jaime has grown quite apathetic to general goodness and honor. Where Cersei is convinced of her own rightness and goodness, Jaime recognizes himself as a morally questionable man who has done morally questionable things. He still walks a sort of line, when it comes to honor--though it's a perilous line. Jaime tosses Bran Stark out a window very early on in the series, but Jaime also refuses to kill Eddard Stark after winning their street duel. Jaime doesn't go out of his way to save smallfolk on the road, but Jaime does turn right around to save Brienne of Tarth from whatever fate she was to be left to at Harrenhal. He largely respects people based on their skill with the sword. Anyone useless can get on the first horse out of town.
Jaime's arrogance is a defining trait, and something that he has earned. The truth is, before getting his hand lopped off, he really was the greatest swordsmen of his time. Friends and foes alike would both cop to that, and Jaime has a list of battles to prove it. The loss of his sword hand has tempered his arrogance and done a great deal to humble him. At one point, Jaime tells Brienne that he was his hand, that without it, he has been reduced to nothing. Brienne does not hesitate to berate him for this complaint, pointing out to Jaime that there are men who have lost much more than a hand and have not given up as Jaime is giving up. She did not, she says, take him for a coward, and this is part of what spurs Jaime back to life--and down a road of redemption.
Before he was anything, Jaime was a warrior. Members of the Kingsguard serve for life and are not permitted titles, lands, or marriage. Jaime forswore his birthright as Tywin's heir when he was knighted, and he continually meets suggestions that he set these vows aside with vehement refusal--even to Cersei, the woman he loves. He lacks the political ambition of his father and his twin--much to Cersei's frustrations. She consistently tries to build Jaime into her schemes, and he generally goes along with things--but it's when she tries to get him politically involved that he outright refuses. Jaime has no interest in becoming Hand of the King or any part of a behind-the-scenes monarch. He isn't even interested in acting as a father to his children. Jaime is a warrior, a soldier, a knight, and he is content to be these things, to deal with his problems more with action and reaction than political maneuvering.
For all the oaths that he has broken, Jaime has always been true to Cersei. Their relationship is deeply important to him, and he loves his sister absolutely. That's not to say that he is blind to her faults, but he loves her for who she is. It's an incestuous love, but a deep and true love. And Jaime's love for Cersei motivates him in a great many things. The thought of returning to her arms spurs him on toward King's Landing; the thought of marrying her in front of Gods and men is something he truly wishes for, to live honestly and as they were meant to live, totally together. It's only when he's told that Cersei has cheated on him that Jaime begins to turn from her. It helps that this revelation comes right around the time Cersei is certifiably losing it. At the end of Season 6, Jaime is living with the realization that his beloved sister has turned into a mad queen right on par with Mad King Aerys himself--and her interest in wildfire only deepens that parallel. Jaime has by this point removed himself (or been removed, depending on who's side you're on) from Cersei's bed, but he still has a sad kind of love for her.
Jaime also loves his brother (for clarity: loves fraternally, platonically, no incest). Jaime is the only living Lannister who has ever treated Tyrion decently, and despite their many differences, the brothers are very close. Jaime admires Tyrion's cleverness and wit, and they happily spend time in one another's company. Jaime would be Tyrion's chosen champion when he is put on trial for the murder of Joffrey, and though Jaime must refuse based on his Kingsguard vows and position in the Lannister household, he refuses with regret. He never comes down firmly on whether or not Tyrion killed Joffrey, though he does seem reluctant to believe that such a crime could be true, and more or less takes Tyrion's word for it--though not so much that he is willing to break vows or risk their father and speak up for him. He does help to free Tyrion and save him from his execution when the trial goes awry. Except then Tyrion murders their father. Not the intended end.
Jaime can put on tact and decorum, but prefers to speak honestly and bluntly. He's much like Tyrion in this respect. Less inclined to disdain and berate people for minutia, like Cersei, Jaime still enjoys mocking people he perceives to be incompetent or inadequate. It's only once they've won his respect (typically by being skilled fighters) that he begins to treat them with real seriousness and decency--though even after that point, he's more likely to treat friends with sarcasm. His respect is not easily earned, but once it has been earned, he keeps it for life, even after those same people come to hate him. The Kingslayer can handle the hate, and handle being deeply misunderstood. He's been handling it for a long time.
POWER:
GOLDEN GHOST HAND. With concentration, Jaime will be able to manifest a gold and ghostly version of his missing right hand. It will take time to level this power up to something useful and manageable, but if he works at it, Jaime will eventually be able to interact with physical objects--most importantly, swords. The hand will maintain a normal human strength.
once he starts practicing, the leveling will go in stages.
{WEEK 1-2:} ghost hand will be visible only to Jaime
{WEEK 3-6:} ghost hand will be visible to others, but will not be able to grasp anything at first. midway through week five, the hand will be able to pick up small objects (think pencils, paperclips, forks) and hold them for a few seconds before the hand loses permanence and the objects are dropped.
{WEEK 7-9:} ghost hand will grow stronger through this period. it will become more visible and obvious, and Jaime will be able to grasp and lift bigger and bigger objects and hold them for greater amounts of time (a minute at first, and then longer, with practice).
{WEEK 10:} ghost hand will be fully useful and will be able to hold a sword without dropping it. He will need to continue practicing to give the hand's manifestation a greater stamina.
If he stops concentrating or is distracted, especially distracted by pain, the ghost hand will do one of two things, and it's a total crapshoot as to which will happen. POSSIBILITY ONE, the ghost hand dissipates, and drops anything that Jaime is wielding or holding. He will need to concentrate very hard to bring it back, and it will be weak and require more effort to sustain. POSSIBILITY TWO, the ghost hand will totally glitch out and morph into a lion's paw, claws and all. Lion version of the hand will be very unstable and feel disconnected to him, like it's controlled by someone else. Lion hand will disappear only if he gets knocked out or falls asleep.
Manifesting the ghost hand will need daily practice. If he stops practicing or gets lazy, the stamina of the hand will degenerate and grow weaker, and he could lose it entirely.
OATHKEEPER. If Jaime swears an oath or makes a vow, he's now bound to keep it by pain of... pain. This power is activated only by oaths or vows that end with "I swear it by the Old Gods and the New", which is the traditional styling of a knight's vow of fealty. After he's said those words, Jaime will be bound to keep his promise. Trying to defy or break the promise will give him a headache. If he tries harder or stretches the bounds of the promise, the headache will get worse, and will prompt nosebleeds, possible ear bleeds. If he really does break his word, he'll die. Medically the cause of death will look closest to an aneurysm, but it's really more serious: ruptured honor.
KINGSLAYER. Jaime will be uncommonly good at chess. Not because he's very clever and good at strategy, but because every path to the king will become immediately clear to him. Once he's learned the rules, he will be able to win a game in only a few moves. Bobby Fischer, eat your heart out.