So we are not given enough information on Kate's past to know if she really murdered her step father, but she might have. She seems to have a good heart, but the marshal that is chasing her does not seem to think so. In the beginning of the series we find out that she was arrested in Australia, and is traveling home handcuffed and all to await trial. Jack and Hurley find the marshal's case file on her past crimes early on and Jack calls her on it. Kate is a very closed person and does not tell Jack enough to know what happened.
We also know that Kate helped her persuaded lover rob a bank in the past. The interesting part here is that she did not rob the bank for the money, but for a little toy plane in a safety deposit box in the bank. We are later led to believe that this plane was her childhood sweetheart's and it had emotional attachment. We are also allowed to find out that she did something to harm her childhood love and that is why the toy is important.
Kate is currently one of the leaders of the camp, and she is invested in trying to leave the island. It may be her need to run away that urges her to leave the island because she does not seem to have much to back to. Whatever happened between her and her step father has caused Kate's mother to hate her. Kate plays a strong role in the lives of other characters on the show as well.
Kate is in a strange sort of love triangle between Jack and Sawyer. It is obvious that they both are in love with her, and it seems that she wants Jack in the beginning. She kisses him at one point when they are alone in the jungle, but then gets scared and apologizes. She later on sleeps with Sawyer, multiple times, and Jack finds out. Jack seems to still love her, but he ends up shacking up with one of the others for some reason.
Kate is very mysterious because she does not let anyone into her past. She refuses to talk about her past to everybody, and this may be one of the fears she has to overcome. In the last episode of the last season she appears. She is in one of the flash forward to the future sequence, and she looks healthy. She is not in jail, and despite Jack wanting to go back to the island, it seems that Kate is happy and doing well.
I do not know what will happen with Kate, and I do not know who she ends up with in the future, but I know she will make it interesting to find out!