We don't see much of Jaime Lannister in A Clash of Kings. He spends most of the novel pulling a Count of Monte Cristo, hanging out in a jail cell at Riverrun and practicing his beard growing. He only appears in Catelyn's final point-of-view chapter, but in that short time, we do get some tantalizing hints about Jaime's past and his character.
While Catelyn and he are playing twenty questions, Catelyn argues he shouldn't call himself a knight since he has forsaken every vow he's proclaimed. To which Jaime replies:
"So many vows… they make you swear and swear. Defend the king. Obey the king. Keep his secrets. Do his bidding. Your life for his. But obey your father. Love your sister. Protect the innocent. Defend the weak. Respect the gods. Obey the laws. It's too much. No matter what you do, you're forsaking one vow or the other." (56.Catelyn.136)
Jaime then relates the story of how King Aerys, the Mad King, killed Ned's father and brother, Lord Rickard and Brandon Stark respectively. The Mad King slowly burned a fully-armored Lord Rickard alive with wildfire, and while Rickard was being burned, Brandon was brought in with a leather cord around his neck and a sword placed in front of him. He was told he could save his father if he could reach the sword, but as Brandon struggled, the cord tightened and strangled him. So. Gruesome.
This is important because it's the first time we've been related a firsthand account of just how crazy the Mad King was. We have to ask ourselves, should Jaime have kept his vow to such a malicious monarch? What vows came into conflict that made Jaime feel it necessary to murder his sovereign lord? Finally, does Jaime deserve the scorn he receives for this act of regicide, being called "Kingslayer" and such?
Since our time with Jaime in this novel is limited, we don't receive any answers to these questions, so instead we'll have to carry them forward with us. The answers will no doubt be worth the wait.