There is very little know about Karai's past. It seems like she was trained to be a kunoichi at a very young age.
Karai is very rebellious, brave, and an expert kunoichi. She is very also seems to have a soft side when it comes to Leo.
Shredder: Karai and her father have a very complicated relationship. They both seem to get on each others' nerves, and don't always agree with each other.
Leonardo: Leo and Karai have a very flirtation relationships, they both seem to have a crush on each other. When they first meet she was trying to get him to break the rules. When fighting the Kraang, Leo notices Karai watching him and his brothers, he seem like he wanted to impress her. In Enemy of My Enemy Leo didn't really want to go against Karai to get Shredder. At the end he seemed really disappointed that he could never get her to be on their side.
Sometimes they place Karai as Shredder's daughter or granddaughter.