Does it mean that you try telling me something about me?
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Your question, 'What am I like?' could have two meanings depending on the intonation.
If you were to ask someone, with sincerity, 'what am I like?', then yes, it means as you suggest, tell me what I am like (something about myself).
If someone has done something silly and asks, 'what am I like?' with emphasis and raised pitch on 'what' and 'like', it then becomes a rhetorical question - more of a statement.