Yes, that is correct. I want to make friends. I want to be friends with her. I hope we become friends. I think she would make a good friend. I like her, and we could become great friends.

"I want to make friends with her" is perfectly fine. You are stating a fact. "I'd like to become friends with her" states a wish. This is polite-sounding, but your original sentence is still grammatically correct (although slightly direct).

i'd like to become friends with her*

It's perfect.