How can i encourage friendship with this girl, then ask her out?

Well it does seem she may like you back. So really talk to her and get to know her, but do it casually, so it doesn't seem your hitting on her or something dumb like that. After a while of talking a lot, she'd be considered your friend, and you could ask her to hang out. Then pop the question if she wants to go out with you, and i don't think she'd say no.