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I have always used "What are you doing", but a basic grammar rule is that for plural things you use "are" and for singular you use "is".
But when you say something like "What are you doing?", you are talking to a singular person. So what's the explanation of using "are" when talking to a singular person?
You use is for third-person singular subjects. The subject here is the second person singular pronoun you. Though subjects normally precede the inflected verb in English, the order of the subject and inflected verb is inverted in certain constructions, such as this question construction.
E.g.,
What do you think about this car? / What does he think about this car?
Have you ever been to China? / Has Alice ever been to China?
The reason for this anomaly is that you is intrinsically plural. It is being used instead of the singular thou and thee out of politeness (or at least that was the original reason - see: https://en.wikipedia.org/wiki/Thou#History).
