Make eye contact and smile at her when something funny comes up in class and everybody's laughing, for instance if the professor or teacher makes a joke and the whole class is laughing, make eye contact with her and smile. It sends her signal that tells her you're into her. Don't do it all the time, or it might seem kinda of stalkerish and once you got her smiling back, start saying hi to her when you walk into class and then maybe you guys can talk after class. If she doesn't smile back the first time thats ok, keep trying about 2 or 3 times and if you still don't get any facial expression back forget it, hoped I helped :)
I think the best way would to be what you said, to catch her eye. Hold her stare for a little bit longer than you normally would with our glazing over or staring too deeply into her eyes. Make sure you are smiling too, otherwise she will get creeped out. Also, just chat with her in class, be funny but not cheesy, and goof around a little with her. Try and find out about her interests (fav. Band, tv show) and bring it up or hum the song around her and she if she lights up. Ask her for help with the class and try to create an opportunity (maybe relating to school or just your class) for you guys to hang out. Even if its like working on your bio projects in the library after class, she's with YOU.
Another idea is to remember small things she says in conversation like her fav. Food or movie and show her later you remember and care by giving her a handful of M&Ms (b/c its her fav. Food) or watch Forrest Gump (because its her fav. Movie). Be friendly, nice, not toooo showy-off, gain her attention/interest in a reasonable way, and smile. :-)
Ask her questions about what your doing, look more at her than the paper and smile a lot when you do that. That makes her feel special and leaves her thinking about you after class. After a while you can start to talk to her randomly. You can say silly things to make her laugh and steal her pen for the class hour. Take her papers and try to get her to grab them from her hands while you pull away. Scribble on her pages of work, when she turns through her book she'll see it and laugh. If she's playful then she'll flirt back by writing on your book or taking your stuff.
You should take it easy , relax a little. And don't get too jealous if you saw her with another guy. Just take things step by step. First you start with saying hi byes to her , then you observe her reaction. Then try to help her along if you see her need some help , eg when she got trouble carrying her books. Thirdly , try to meet her in school for something that is worth the excuse. Eg , a meal or homework. Lastly , you ask her if she is free. I just don't want you to do some crazy stuff , I saw it once on the news a couple of week ago that makes me sick. Some dude was drunk , he drove his car over the bridge and had his girlfriend in the trunk and she was pregnant with his kid. Deep inside they found a tape , but they didn't say who he was. Come to think about it , his name was. It was you. Damn.