Visa's How To Approach A Girl 101
After you've done all the subtle flirting and eye-contact stuff, stare at your phone for a while and pretend to be doing some important messaging. (If you really want this to be especially effective, actually get into a heated, opinionated text-argument with a friend- you can show her the messages yourself, which gives you some opportunity to get physically close to her as well) At some point in time, give an especially exasperated sigh, roll your eyes- something that'll make her notice that you're annoyed with something on your phone. Give her a "tired" i-don't-know-why-i-go-through-this sort of smile, and she'll most probably smile back.
Ask for her opinion on something. You can't go wrong there. Don't ask for the time, because that's close-ended. Go over, give your most friendly and charming smile (again) and say something along the lines of "sorry to bother you, but i'm in a heated argument and I really need an opinion..." Nobody's going to turn down giving someone an opinion- not even the hottest of girls. If you seem like you genuinely want their opinion, they will most certainly give it to you- it's a small and simple favour that takes no effort to do. Try to ask a question that's complex and needs some thinking. Observe her in advance and try to make it relevant- is she reading a book? is she wearing glasses? is she wearing shoes? you could be pretending to argue with your friend who claims that the girl he likes is going to think he's a creep if he buys her lingerie, but you think it's kinda sweet. Whatever. Do guys cheat on girls more, or vice versa? Who lies more in relationships? Something subjective. Asking a funny but interesting and effective question is an art form in itself. (Pickup artists call this the opener, and often have many different ones at their fingertips.)
Along the way you end up progressing into a normal conversation, with an escape clause if there isn't any chemistry. Have your body language directed away initially so that she doesn't get uncomfortable with your sudden invasion of her personal space- sit at the side of the seat opposite her, for example, and be on your toes as if you're about to leave any second. If she gives you more attention, slowly and casually move more fully into the seat- and before you know it you're on an impromptu date. Say, have you ever tried the donuts down the street? They're awesome!...
You're welcome!