How do I approach a girl who hates me?

First of, you need to know that is a harder to impress a person that hates you than impressing someone that likes you. Here are some tips you want to try.

Discover why she hates you, if it is a bad habit that you have that she hates, work on it to improve yourself. If she sees a change in you she might have a change of mind. Also, having a conversation with her might be of great help, it might be that she misunderstands you, having a conversation can help you.

Be friends with her friends. They will help you know how best to impress her and make her like you.
Find out what she likes and try to be good at it, this will make her notice you.

If you try all these and there is no change, I advice you stay away from her, she might notice your absence and come looking for you.

It is difficult to approach anyone that hates you. I would start by understanding why they hate you. It would require a little bit more than just knowing why. You should try to be sympathetic with how they felt about whatever it is that you did. It could be something very small and they were already having a bad day.

That small action would be similar to adding additional baking soda to an already erupting volcano. I would also say you should give it time. Time is the ultimate healer. With enough time, the person would either forget what you did or it would no longer be as significant. Also, if you add good deeds to the mix, like getting her what she likes, especially without her asking, or being there for her, it would help her see that you are not defined by that one occurrence. Good luck.