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 :)

friend,I don't know your name so I call you my friend.
I guess it is your first time to try geting closer with some guy initially.So I understand how hard it is to make a first step.
friendship is based on equality of two friends.But now you feel so shy to get closer which says you are not so confident for this relationship.why?Because you keep thinking I need to do something to make others like me.And that is wrong.
There is nothing you need to do.But if you need more friends,then spend less time in the library or in home,trying to just be with those guys and spend some time together,you just show yourself and be yourself.somebody will like your way of life,your personality etc.then it is easier to make friends with them.Actually some will come and make friends with you instead.
Confidence!If you shy away from anything,then anything will shy away from you.have you learned about the theory of Newton?