"Do I know you?"
Translation:Czy ja cię znam?
You can remember, as a general rule, that a pronoun follows a verb:
Znam cię = I know you.
When a sentence has more words, it's natural to move it before a verb:
W ogóle cię nie znam = I don't know you at all.
If you want to stress a pronoun in some complex sentence, you use its longer form:
Nie znam ciebie, ale jego tak = I don't know you, but I know him.
You can also put it at the beginning. The stress is even stronger here:
Ciebie nie znam, ale jego tak.
Hope this helps.
Frankly, I think that "cię" is better here. "ciebie" is an emphasized form of the pronoun, so it's usually used when you need to contrast it with someone else or after a preposition.
As for "Czy"... well, you can omit it basically everywhere, but you shouldn't start a question (nor any other sentence) with a pronoun other than the subject pronoun. So "Czy ja cię znam?" could also be e.g. "Znam cię?" but not "Cię znam?".