After the birth of Joffrey, Cersei realized if she wanted her children to pass for Robert's, she would have to bare him a legitimate child. She never thought she would love her child as much as she does. Somehow, Lina grew to hate Jaime, after the birth of Myrcella and Tommen. She knew she was different. With this knowledge, Lina seemed to latch on to Tyrion, whom she adored like her younger siblings. Lina grew to be her mother's protégé.
She's wise beyond her years, thanks to the teaching of Varys. She gets her cunning from her mother, her looks from Robert, her sword skills from Jaime and her cool head from Tyrion. Her first kill happened when she was 9 years old. She was lost in Flea Bottom, and someone tried to rape her. She cut his throat and ran away. She returned to her home and never spoke of this.
When she was 10, Tyrion recieved word from Robb Stark's army that if they did not send Lina, Robb would take the city and rape her himself. So, with everything she needed, she left for Riverrun. No one told her she would be married off. Cersei had orginally sent Osney Kettleback, but Tyrion sent Bronn along after she left the gates. When she asked how Bronn knew Tyrion, he told her was simply his sellsword. When they finally reach Riverrun, Lina is having second thoughts, but knows she can't back out. King's Landing depended on this marriage.
When Lina finally meets Bran, She's surprised to see he is not completely depressed, due to his crippeled legs. It would be difficult for their futured life.