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Questions & Answers about Kennst du mich?
Why is it mich and not ich here?
mich is the accusative (direct object) form of ich ("I"), just like "me" is in English. Because you are the person being known, you receive the action of the verb, so you must use the accusative pronoun mich.
I thought the verb for "to know" was wissen. Why are we using kennst here?
German has two common verbs for "to know". You use wissen when you know a fact. You use kennen when you are familiar with a person, place, or thing. Since you are asking if someone is familiar with you (a person), you have to use a form of kennen.
Why does the sentence start with the verb kennst? Where is the "do" from the English translation?
German doesn't use helper words like "do" or "does" to ask questions. To turn a statement like Du kennst mich ("You know me") into a yes/no question, you simply swap the verb and the subject to get Kennst du mich? (literally: "Know you me?").
How would I ask this formally, like to a stranger or a boss?
You would swap the informal du for the formal Sie, and use the matching verb form ending in -en. The formal question becomes: Kennen Sie mich?