"Don't I know you" vs. "do I know you"

Negative questions are used to express surprise and when we expect agreement from the listener.

Q: Don't I know you? >>> Expected answer: Yes, we've met.

(This is similar to "Haven't we met somewhere before?")

On the other hand, positive questions are really seeking information.

Ex. Why do you keep staring at me. Do I know you?

Finally, you bought tickets to a concert, you know your friend's crazy about the band but she doesn't seem so excited.

Surprised, you ask: "Don't you wanna go?"

Do I know you?

suggests that you think you don't know the person; you are expressing doubt that you know him.

Don't I know you?

suggests that you think you know the person; you are expressing doubt that you don't know him.

The answers above are good. One thing I should mention though is that they most definitely aren't equally polite.

"Don't I know you?" is useful as a way of opening a conversation, as there can now be a (sometimes long) discussion about various things you may have in common. Even if the answer ends up being "no, you don't", you actually will know each other a little by the time the process is done.

By contrast, "Do I know you?" is often used to shut down conversation when someone you don't want to talk to tries to strike up a conversation. Its kind of a rude way to do it, but some folks consider just walking up and talking to someone without an introduction to be rude as well...

"Don't I know you?" is used in puzzlement when one is unsure of the "you"'s identity (perhaps it is someone whom one has met before) and indicates a willingness to increase interaction, whereas "Do I know you?" indicates that "you" is someone with whom there seems to be no reason to continue interaction. The latter is challenging: provide your bona fides or get lost, the former more friendly.