In my experience, after class is one of the worst places to be trying to introduce yourself to a girl.
First, she might have other things on her mind than chatting with strangers. Remembering the lecture, getting to her next class, etc. You'll feel embarrassed if you try talking to her and it doesn't seem like she's all there.
Second, and this especially applies for a morning class, some people just do not give a crap if it's their only class in the morning. They roll out of bed, go to class, and then go straight back to bed. This means she might not be looking/feeling at her best and many people tend to shut down when they're feeling self-conscious.
Third, it makes it look to your professor like you're going to class to hit on girls, or that you're otherwise more interested in the pretty girl a few seats away than the lecture. If there were 200 students in your lecture, he might not notice. 30, and he's going to be able to perfectly tell who's paying attention and who isn't. If you're a freshman, you want to start getting the professors to like you much more than you should be worried about getting pretty girls to like you. College is full of pretty girls, but you're only in that class 45 minutes every other day. And, trust me, the girls you want anything to do with are the ones who like guys who are more interested in getting things done - be it classes, school clubs, what have you - than guys who seem to only be focused on women.
And lastly, you have no "in" with her. By that I mean you have no starting point for a conversation or let alone a relationship just by being in the same class. Trying to leverage that by making small talk about a class she may or may not be interested in will not work if you're not interested in the class either. Plus, people can smell duplicity like that a mile off.
The "secret" to dating in college, if there can really be said to be one, is to make yourself an interesting person and let women come into your life through the things you do rather than build up your life around deliberately trying to get girls to like you. Unless all you're looking for is noncommittal one-night stands, then all that strategy is going to get you is loneliness.
Aramis Night said:
Besides. Depending on the state your in, it may be better to ignore her entirely to avoid getting caught up in anything that could lead to you getting kicked out of college. As has been demonstrated recently, all it takes is an accusation of wrong doing and your college life will be over. Don't hand that kind of power over you to anyone. Plenty of time for dating when college is over.
Well, that's a little extreme. I wouldn't waste time being overly worried that she's going to rope you into some kind of bizarre rape accusation scheme.
Now, that said, some college girls are genuinely batshit insane (albeit of a much more mundane and less dramatic sort) and you do need to protect yourself. But that's no more difficult than not drinking with her until you trust her and know how well you handle alcohol, using protection, and not getting physical or otherwise serious until you can be sure she's not the "clingy" serial monogamist sort who is just trying to corral you into a relationship.