I usually start with jokes or comments.
I hang near the person and wait for something to happen, usually someone else doing something stupid, and I make a humorous/sarcastic comment. Something that expresses an honest opinion (ex.: someone talks about how guns are great and I say "yeah, I await the day we can all carry our own personal murder weapons"). If the person shares my opinions they laugh, agree, or at very best join in, and either way I have an "in".
So in one comment I can judge whether someone shares my opinions and enjoys my sense of humor while beginning a conversation. From there it's just talking.
Sarcasm is my tool of choice, may it never fail me!
Edit: The above is to meet someone. If I decide to ask out someone I've been talking to for a while I like to ask at the most random time, in the middle of conversation, absolutely casually. No that's not a joke, I swear. It works:
1) It catches her so off guard you immediately see her real reaction.
2) You have just been so absurdly brash and confident that surely she should say yes, if even just to see what you'll do next.
3) It's just damn fun!