Keep up conversation with her, and eventually ask "Hey, if you're not doing anything this weekend, wanna hang out"? Take the whole boyfriend/liking slowly with her, and you can also slip in comments along the way, should that topic come up about "Yeah, my mother was saying the same thing about me finding a girlfriend"

The important thing is to keep it casual, be confident, and be relaxed, but, for lack of a better word, "flirty"... Also, IRL is far better than any kind of IM, so even if you call her to ask (and say "I haven't been on IM lately, I've been working, so I wanted to catch you this way, I hope you weren't busy"), it's again, far better, since even if you're not there, your voice conveys far more emotion & inflection than IM ever could hope to; Avoid IM like the plague until you're more established with her, if at all possible;