International journals select/build their reviewers panel in the following ways.
1. If an author has published with the journal, the editors may request them to review a paper that aligns with his/her expertise. Therefore, publish in high quality journals that are indexed in web of science. Publishing in such journals can help in serving as reviewer for them.
2. Some journals are always actively seeking reviewers, so look out for those journals in your discipline. Look up the website of your favourite for call for reviewers. Send a brief cover letter to the editor indicating your interest to review along with your updated CV.
3. Join Publons academy for a free course about peer review and the mentors might be able to help you in reviewing for a journal.
4. Some journals also offer peer mentoring programs to prepare novice reviewers. Look.out for those journals.
A very good answer by Ahtisham Younas. I used to review articles for the Mark Allen Group (MA Healthcare). I had submitted a few articles to them which they then published. They then commissioned articles about various topics and after this, asked me if I would carry out peer reviews.
This was for nursing journals and not your field, but I hope you are soon successful in finding a journal to review for. I, myself, learnt a lot from reviewing and although the articles were in my own field, I used to read further if I was unsure of facts in the article to be reviewed.
Ask your supervisor, classmates, and colleagues because some of them may serve on editorial boards of journals and/or conferences. Start with a few personal contacts. Once you get into the system as a reviewer, they will never leave you alone :). I get around 2 requests per week :( - have to decline most of them. Also note that most journals rank the reviewers based on the quality of their reviews, so do a good job!
This question seems to require a discipline specific answer, as well as an editor specific answer. Each is a little different. One rule of thumb, however, is that you usually have to be published before you can start reviewing for a journal. You don't have to have published in the journal you wish to review for, but not many reputable journals are going to place someone without publishing experience in a position to judge the work of others for possible publication.
One way that I recommend is when at a conference to go up and introduce yourself to a journal editor, share your card, and offer to review if needed. You can do this be email as well. A variation of this is to send a congratulatory email to a newly appointed editor and offer to review then. Keep in mind what Alexander Serenko wrote, and be careful what you for.... Good luck.