Great question! Yes, we can use "gonna" to say "going to". You will hear this frequently by both American and British English speakers. Using "gonna" correctly will help your English sound more natural. "Gonna" is not rude or bad English. It is just the shortening of a phrase that English speakers use very often. In fact, the "What are you" part of your question is often shortened as well. And so the question, when spoken, can be shortened from "What are you going to do?" to "Whaddya gonna do?" When writing, and especially when writing formally (for example, a job application cover letter, or business correspondence), yes, we should write out the full "going to". But when spoken, whether it's a family member of the President, you will hear "gonna".

Xarmanla, don't be so angry girl!))

I don't think so, Peachey. but thanks anyway.

The thing with slang is that you cannot just throw it into a sentence and expect to sound "natural". In fact, you'll sound very, very unnatural. For your example sentence, the pretty grammar construction at the beginning makes "gonna" stand out like a sore thumb. So your options are "What you gonna do?" or "What're you gonna do?" Really, there's nothing wrong with "What are you going to do?", so saying the sentence properly should be your first option.

I disagree with ESL Teacher Hamish. I do not use the word and no one I know uses it either.