Same here. I contacted an editor of a journal (IF=4) requesting to be added to the reviewer database but he replied saying that one needs at least 10 years of research experience along with two references of established authorities to be accepted as a reviewer. My first break came when some author recommended me as a reviewer and things took off from there.
I have a feeling that if you do a good job reviewing for one journal, word gets around and you get more invitations. Sure, reviewing is hard and takes up lots of time but it has its own rewards. I have never read so many papers (remember you gotta read the references to know what they are talking about) in my life before.
Here is what I would recommend to anyone who wants to review for a Journal:
1. Write to the editor expressing your desire to be added as a reviewer. Once you are in the database and you choose the subject areas you want to review, you automatically start getting reviews.
2. Understand that reviewing is tough. It takes up time and resources. Also, just like any other endeavor in life, you can only continue reviewing as long as you do a good job. Editors tend to drop reviewers if they feel they are not getting good reviews. So, be prepared before you start approaching the editors.
3. Request your friends that are already established to forward the 'request to review'(s) that they are getting. You advisor is a good person to start with. Once you've moved on from the university you graduated from, consider your advisor as an ex-colleague who likes to look after you. This is one other way towards getting that first request to review.
4. Make a point to talk to editors/associate editors during meetings or conferences. Pass them your card (sometimes your resume, even) and follow up with an e-mail requesting them to consider you as a manuscript referee.
