This is easy Tedward. Don't progress from "hanging-out" to dating. Go straight to dating, that's what you want to do anyways, why hide a date behind the cute non-aggressive "hang-out" session.
Here's something that chicks dig. Chicks dig a guy that they can't figure out. One that she isn't sure likes her or not. As silly as it sounds, it's the way it is in the dating world. You want what you can't have. And what you aren't sure you can't have, can just as easily be yours with some work. Hopefully, you're still following me.
Stop talking to her on AIM. Become a challenge. Next time you see her in person ask her "Hey, it was nice talking to you, what's your home phone number?" Since you're still okay acquaintances, then this should not be a problem. You get the number, don't make any contact with her, AIM or otherwise, better yet, block her for a few days, but just be sure she can't find out that you did. Call her up, 4-7 days later and say
"Hey, I'm free <insert day>, let's go get a bite to eat at <insert food joint> at <insert time>."
Your terms, no questions, simple reject or accept. If she accepts, you've got yourself a date. If she rejects with NO COUNTER-OFFER (different date), then you are in a boat full of no romantic interest. This is the point where you give up since there will never be anything between you two.
But for explanation sake, let's say you get on the date. Have a good time, don't be nervous. She's lucky you could find time in your busy schedule to go on a date with her. If you feel that the date went well (ie: you both had fun), then KISS HER.
You MUST kiss her. This is the absolutely only low-risk way to find out if she is interested in you. All while making you look bold and confident. If she cheeks you or pushes you away, that is a sign of extreme disinterest. If she accepts, you're in. Go home, smile, and try to ease of the excitement and go to sleep. Make it all seem natural to you.
If this all seems a little much for you (which I bet it does), sit down for a second and think what you want from this. You want a relationship. What you don't want is a relationship with a disinterested girl. That is why you ask for the number, that is why you ask for the date, that is why you kiss her. You display your interested and if she has ANY romantic interest, she will comply. If she doesn't, why go out with her at all? Don't waste your time figuring out if she wants to be with you. Find out the classic way without all the emotional fuss.