How to make friends in university and become closer to them?

How to make friends in university and become closer to them?
I have issues of being alone at uni.I meet people occasionally,but as soon as that class is finished,we don't really meet up or extend anything.Know what i mean?I think my issue seems to be mainly from conversion i just cant seem to really get into conversation with people and because of that i shy away from any real extracurricular encounters which i know helps to solidify friendships.I spend most of my non-class uni time in the library and i leave as soon as possible or in my residence.
I just met a British girl who is really nice and friendly,we take the same course,the same tutorial,even join the same society.I couldn't have the chance to get to know her since I saw her in the first year ( It's already my second year now!) I'm sure that she is exactly the kind of friend I want,so do you have any ideas about how I can get closer to her and become good friends?
I feel a bit awkward to sit just next to her,that makes me feeling like I'm desperate to become her friend.:)
Also how to get people's number when you become a new friend with them?
We just knew each other yesterday and she also has two real friends too...but I still felt shy when I sat next to her in the lecture today :)