How to become a peer-reviewer?
As an editor, when I need peer-reviewers, if I don't know someone with expertise in the topic at hand, I look to see who has published peer-reviewed work in a similar area recently. No-one signs up to do reviews, it's up to me to go out and find them. Which means, if you haven't published peer-reviewed work recently, you aren't likely to get any invitations.

So, in my case, if you want to do peer-reviews, the first step is publishing something in a peer-reviewed journal. Otherwise, no one will know you're there.

This doesn't mean someone who isn't publishing is never qualified to provide a peer review. But if I don't know you, and you haven't published anything yourself, I have no way to evaluate if you are qualified to provide a good review. 

Another thing to keep in mind is that the distinction between an interesting paper, and an interesting publishable paper, may not be clear to someone who isn't actively pursuing academic work in the area.