Just start a conversation with her about something you notice, that you may share as a common interest, then introduce your self. If nothing else say something about the class. I think you'll figure some thing, no advice is universal, and heck remember not ever here may give the best advice. Although I am one to talk, my last relationship didn't exactly work out.
Just sitting next to her is not enough if you're going to be stumbling over every word every time you try to say anything. You need to have something to say. And that means jokes.
You start off by sitting near her while pretending to be talking to someone else who just happens to be near her. You start telling jokes. If you don't know any, get some. If she's wearing headphones and you're telling good jokes she will notice everyone around her suddenly laughing. She will take off the headphones to see what's so funny.
You always need to leave them wanting. So tell a few jokes and then get about halfway into something really juicy and naughty. Then say, "Oh wait - I gotta go. Sorry guys." As you're leaving, everyone will be disappointed, say "Awwww..." and stuff like that. Here's what she will percieve:
a. Everyone is having a great time as soon as you show up.
b. Everyone is miserable as soon as you leave.
She wants to feel good, have fun, and all that. The next time she's wondering what she can do to feel better, she will think of you.
Now for the bonus round, you're going to have to convert the general good feeling into specific feelings toward you. If you ever find yourself alone with her you can start off with a joke but then transition to something dramatic and "deep". Like you talk about the rain forest being destroyed and end it with "I dunno..." You're not offering any actual solutions to the problem (which can be analyzed and refuted) but instead you're signifying how concerned you are. Find a few things you're concerned about like this and let her know that you're concerned about our future as a planet when things like this can happen.
Here's the key to this last one. Guys think in terms of problem resolution. The rain forest is being destroyed? Why not try replanting it? That ought to fix it. But girls don't think in terms of problem resolution. They think in terms of expression and sympathy. The rain forest is being destroyed? Oh that's so sad. How does it make you feel?
Eventually, you will want to get closer and closer to this girl. Here's how you seal the deal. You stare into her eyes for what seems to be far too long and she gets suspicious and maybe a little bit weirded out. Out of the blue you just laugh. She asks what it is. You tell her it's nothing. Now she HAS to know what it is. C'mon tell me. What is it? That's when you tell her that even with all the horrible things going on in the world, every time you're with her everything just seems to get better somehow. You just feel good whenever you see her.
The key to this last one is that you have already engineered it so that she already feels great when you arrive and miserable when you leave (see above). And then you tell her that you feel the same about her, but for an entirely different reason.
Right about then, a little voice will whisper in her ears, "Soul mates!"
Then years later when you're explaining to your kids how you met their mother, you can explain how DrGadget brought you two together.
DRGadget's advice is so full of win your gauranteed that girl dude damn wat a silver tongued devil he is, im not being a smart *** im serious here.
That quote sounds like it should go on a book sleeve
In all seriousness though, just start off with a "Is this seat taken" then just make convo about class. Like..."How do you like this class so far" "What's your major" "How long have you been going here" (I assume you're in college, and if you're not then refer back to my previous reply) etc etc, and then say "Oh whats your name by the way, I'm John Connor" (First name I could think of lol)
Watch her for reactions, see if she tries to make convo with you too, asking about you and stuff instead of you doing all the talking, because if she thinks you're cute, she'll try and make convo. Try and sit next to her every class session. Try and get her to look at you when you're walking up the aisles to the seat, smile at her then take a seat next to her. Again, it's all about reactions and body language, if she smiles back, does she smile and look away immediatly, does she say anything when you sit next to her, etc, etc.
I've got a few cute girls in one of my classes this semester, which is a first. Might gonna have to do this russian roulette style because I'm pretty indecisive and I've got a lot of options.