Getting close to a girl can be difficult. It depends how you want to get close to her. This is one of many ways of getting close to a girl to become their friend. Take each of your steps slowly or she will think she is being rushed into things.

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[edit] Steps

Start off by introducing yourself, then ask their name. Do not boast, this can be a really big conversation disaster. Smile at her when you first meet her.
Be careful about talking for too long about the same thing. Sometimes it's good to change the subject every few minutes, but some girls will be frustrated by this or think you're acting strangely. Gauge it by how interested she seems in the current subject. Keep your eye contact, and smile regularly at funny parts.
Never tell lies to her. Within reason. Important things, obviously, you need to be honest about. But if she says "what are you thinking?" and you were thinking that the Italian you two just ate gave her terrible garlic breath, you're better off not mentioning that.
If she talks about a problem or something that bothers her, pay attention and listen actively. Ask follow up questions in order to try to make her see the problem from different perspectives. Generally when ladies speak about a problem is because to ease the burden from their shoulders instead of solving it (unless they explicitly say so, of course). As a bonus, you'll know what she'll dislike.
Show her that you are always there for her, this will tell her you are reliable.
Compliment the girl when in conversation. Almost all girls love compliments about their hair, so that's a safe option. You'll probably get more bonus points for more creative compliments, especially if you think she hears compliments often. For example, "You have gorgeous hair" is nice, but "you have the cutest freckles" will leave more of an impression. Of course, never tell a girl a compliment unless you mean what you're saying.
No matter what try to be humorous, this makes most people more relaxed. You don't need to be a comedian, just add it humor wherever it seems appropriate. This does not mean you need to have funny one-liners memorized: work in whatever humor suits you. That could be amusing anecdotes, sarcastic/wry observations, clever puns, whatever you're best at.

[edit] Tips

This article is how to become friends with a girl as a boy OR a girl, NOT how to seduce one.
Look at some related wikiHows to improve your conversation, appearance and confidence
Try talking to her first and see how she reacts. If she totally ignores you, then she's not interested