According to WordReference, a story is a fictional tale. So to say “mon histoire”, I’d use “My history”.
Histoire encompass both history and story meanings (which slightly overlap), depending on the context.
Mon histoire usually means “The story of my life” which looks to be what you are looking for.
Mon historique is odd, it usually means: “my logs”.
Good answers posted already, I just wanted to hint at the possible use of
Even if not literal, in the case of a person it might be a better option, more humane and less formal/pretentious/academic.