The reason it is especially hard for you is because this is a relatively new experience. In order to become good at (and comfortable with) approaching girls, you are just going to have to man up and do it. Just walk up to her, and say hello, tell her your name, and ask hers. Ask her how she likes the class you are in together... make chit chat about her other classes, or whatever. Then ask her if she wants to grab a cup of coffee with you some time. Keep the first conversation relatively short.
Relax... I'll need some information first. Just the basic facts - can you show me where it hurts?