Close, but "immerse" as a verb is transitive verb needing an object, which is you in this case. It's a good word use here, though. I would say "When I read a book, I become so deeply immersed in it that it's as if I live the lives of its characters".

When I read a book, I immerse MYSELF into it so deeply that it's like I live the characters' lives.

Id use get if I was saying the sentence out aloud but if it's written I'd use become, it's a more proper way I think

Thank you, Sam. Does the verb "get" work instead of "become" in your sentence?

Try this, "when I read a book, I become so deeply immersed in it that it is as if I'm living the lives of the characters." :)