How does a Puerto Rican become Mexican? What’s the process? How long does it take? Can they work in Mexico as actor?

Feeling Mexican doesn't make one a Mexican citizen. This Puerto Rican person would have to have the skills or familial connections to quality to live in Mexico, that's all. This person could also expect to be looked at with suspicion as Mexicans in general don't tend to like Puerto Ricans.

Every country has tough immigration rules, if you want to be a permanent resident or citizen of another country. But if you find work there as an actor, your employer might help with the red tape and forms that need filing.

I suggest you merely "self identify" as Mexican. That kind of thing is all the rage these days.