How can I increase the chance of receiving review requests? Is it appropriate to contact editors to offer my services as reviewer?
I think being a reviewer can round out your overall scholarly portfolio, but that is definitely not the big ticket item for a CV, especially if you are on or seeking a tenure track position.  If you are an early career investigator, focus on publishing high-quality articles, and then the request for reviews will start to flow.  You can reach out to editors and offer your services.  But, to be frank, I would invest your time in producing, rather than reviewing, scholarly works.