I like a girl so much, but she doesn't like me due to my behaviour in past. I realize my mistake and want to make her my friend. Should I pursue her?

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Most people are pragmatic enough to ignore the past in favor of the things they’re deriving from present interactions. Just act like the events in the past never happened. That’s usually all that’s needed. You may need to persuade them you’ve changed, there’s no magic to it, you just say that you’re not the same person, and she either believes you or she doesn’t. If she doesn’t, ask why, and take the feedback seriously. Come back later when you’ve processed it all and don’t think you have the same flaw.