I would only say #1. The other two are certainly not idiomatic, not to say incorrect.
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deadrat"character in the show" is applicable to any performance. "Character on the show" is applicable to TV." "Character of the show" means the nature of the show itself.Right. I should have added that I was only referring to the context given by the OP.
acostaIs "character on the show" applicable to a TV series or a film?I'd just use 'on' for TV, either a single show or a series, but not for a film. "character in a film"