First step (and it's a big one) grow some balls. Girls like masculine men who aren't shy and will go out of their way to talk to them. If you act like something is awkward or weird, it is. If you act like it's cool and natural, it is. So be cool, and be masculine.

Here's a good way to approach her. The next time you two lock eyes, stop and approach her. Tell her, "Excuse me, but, I'm pretty sure you've been stalking me." Make sure you say this with a straight face, and keep it up throughout the conversation. Give her a chance to respond or defend herself, then cite the window example and how she's always where you are in the hallways. Then say, "What's your name, in case I have to file a restraining order?" Now you know her name, and cracked a joke to throw her off. You're using this joke about stalking to make a connection.

Now you say (still with a straight face) "I'm sure you know my name already since you've been stalking me." Most likely, she won't, so tell her. Now she knows your name, and she's going to be thinking about this guy who called her a stalker all day.

Say Hi to her from now on in the hallways and make small talk when you get a moment together. You can keep up the stalker jokes for a little while longer but phase it out eventually. When you're ready (sooner rather later) say, "I think you're pretty cool, so why don't we hang out? What's your number?"