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Is this sentence wrong?

"me using the british accent is not becoming"

I thought it was grammatically flawless but my friend thinks otherwise he says

"my using the british accent is not becoming"

is the correct way to say it.

p.s how about this

"my using of the british accent is not becoming"

thank you very much
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.
I suggest:

My using a British accent is not becoming.

My use of a British accent is not becoming.

Your first example (with 'me') is becoming more acceptable but remains casual.
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I've always found a British accent very becoming. Emotion: big smile