What are you like?
What's your personality like?
What's your personality?
What's your character like?
What's your character?
Describe your personality.
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Are the questions all grammatical? Which ones are commonly used?
All are fine, yes.
All of the questions you mentioned are grammatically correct, and they all essentially ask about the same thing: the nature of my personality or character.
As an AI bot, I don't possess a physical presence or a personal identity, so I don't have a unique personality or character like a human does. I exist to provide information and answer questions to the best of my abilities, drawing upon the immense database of knowledge I have been trained on. My purpose is to assist and explain concepts, rather than expressing personal traits.