Jaime Lannister's Entry in The Book of Brothers (The White Book):
"Squired for Barristan Selmy against the Kingswood Outlaws. Knighted and named to the Kingsguard in his sixteenth year for valor in the field. At the Sack of King's Landing murdered his king, Aerys the second, at the foot of the Iron Throne. Pardoned by King Robert Baratheon. Thereafter known as the Kingslayer."
Sadly, this loose thread from Season 4 will never be resolved. The White Book was probably lost due to Daenerys's burning of King's Landing.
What were Jaime's great deeds after acquiring the Kingslayer name?
Saved half a million lives by preventing the Mad King Aerys from burning King's Landing with wildfire. Sacrificed his sword hand to save Brienne from rape by Locke's men. Saved Brienne from the bear. Gave Brienne Oathkeeper. Knighted Sir Brienne of Tarth. Slayed Euron Greyjoy.