This is kinda long, but here are some of my tips:

1. Be interested in the girl. Sounds so very simple, right? Yet, I find that a lot of guys love to drone on and on about themselves and will never ask the girl a question. Even worse is when the girl is trying to tell a story about herself, and the guy turns it back to him before she finishes. Very annoying and self-absorbed. So, find out what she likes to do, where she's from, food she likes - anything that gets her talking. If you find that she enjoys talking about a certain topic, ask more questions about it. If you can relate to anything she says, share your story, your views, etc., but try not to go off on a 20 minute tangent about yourself. Then, you'll be the perfect balance of listener and interesting conversationalist.

Also, I think this is a great strategy for someone that is socially awkward because if need be, you can let her do most of the talking. And if you are worried about an awkward silence, try to think of a question while she's talking so that after she's finished, you can seamlessly continue the flow of the conversation. This is tricky at first, because you have to listen to her and not faze out to think of the next point of conversation. But, if you get it right, it'll make you look smooth and confident.

2. If you approach a girl in a bar, you have to appear as if you have no ulterior motives. Try not to use a line. Walk up next to her and say something neutral, nothing suggesting that you might be interested, because the girl may immediately put up her defenses, assuming you're out on a mission to pick up chicks. Comment on whatever game is playing on the TV if she's watching, or whatever music is playing. Try to keep it casual.

3. Be conscious of the faces you make. Look at her eyes when she's talking to you. May feel awkward at first, but it'll show that you are interested in what she has to say. Lots of guys' eyes move towards the lips, which is very noticeable and suggestive, or even worse, the eyes dart to the boobs, which is EXTREMELY NOTICEABLE, even for a split second!! Don't think we don't see it. Just because you can't see our eyes at that moment, doesn't mean that we can't see yours.

Also, if she is telling a story and laughing about it, smile or laugh with her. It'll make her more comfortable. Try not to look too serious or intense. This can be mistaken for disapproval or condescension. A few nods, and "uh-huh"s works well to convey interest. Please use sparingly though, because too many looks like you've just zoned out.

4. Try to keep the conversation positive and light. No girl wants to date a complainer or a downer. Not fun.

Then, after you two have had a lovely, interesting conversation, you can ask her "hey, do you want to get coffee some time?" or "can i call you sometime?"

Hope this helps! Good luck!