Nick has an article floating around somewhere that talks about dating co-workers. It is possible to date a co-worker, BUT you also need to knew that there is no magic way to do it. You need to consider the possibility that dating this person may make your professional life difficult. If you don't see yourself working at this job for the next couple of years, then definitely go for it.
If you like this girl, go for it. Her avoiding eye-contact might be a bad sign, but it could mean anything. Your goal as the guy is to show your interest. Invite her out to coffee, lunch, dinner, or other social event. Tell her that you like her dress.
When you ask her out, you will probably get something like a "maybe," or "I'm really tired that night of the week", or something else that is not a yes, but is not a firm no either. At this point don't give up. Ask once more, but add a bit of humor to it. Say, "Oh, I'm a party animal. I'll keep you awake." If she tells you an evasive answer again, probably better to let it go. If she gives you a firm no, also let it go.
THIS PART IS IMPORTANT: If you hang out with her and you are not at work, you can say whatever you want. You can call her beautiful, etc. When you're at work, be very careful before you call her pretty. In America, if I called a girl pretty at work, then someone could take offense and I might get in some trouble. So at work, say things like, "that is a pretty necklace" or "that is a pretty dress."
Also hit on a ton of other girls! It's good for your heart.