Gaining reviewer experience

Strangely enough the simplest answer is, just ask. 

Professors often have a number of reviews on their desk and many are willing to co-review with students.  My advisor in grad school made a point of asking all of us students to review an article with her.  Usually the professor will clear this with the editor of the journal and then submit a co-authored review.  I have done this with a number of students that I have now worked with.  It gives the student a line on the vitae and also experience in how to approach reviewing a journal article. 

I also know of a few journals that have a formal mentorship process set up.  They will match a student with an experienced reviewer so the student can learn from someone who the editor considers a good reviewer.  I think this is a great idea.  Sadly, similar to teaching, students are often just thrown into the process with very little guidance.  Setting up some sort of formal mentorship would probably make the reviewing process stronger, and help train the next group of reviewers to be better than us.

I do reviewing for several student journals--depending on your field, you might have some student-run journals you can participate in as a reviewer or in another capacity. Most journals I've worked with (student journals and reuglar ol' journals) are very open to having new reviewers come on board--if you e-mail them and express interest it's unlikely you'll be turned down. Just make sure to include your areas of expertise and previous experience in the message (or attach a short CV).

I would definitely recommend asking all the professors in your department if they sit on any (relevant to your interests) journals – and if so, ask them to add your name to the journal's list of reviewers, and to let you know if any editorial board or review board positions come available.

Often journals will have a group of associate editors or an editorial board who help find and manage reviewers, and a willing graduate student can be a helpful addition somewhere in that process (which varies widely by journal: some have a dedicated gradiate student editorial board, others include them as trainee reviewers, others incorporate graduate students on an even more ad hoc basis).

My supervising professor for my Ph.D. asked me to do some reviews while I was still a student. He has continued to ask for some journals either the one for which he is an associate editor or for other juurnals who request him to do reviews. I have also been asked independently by other journals to do reviews as well as for conferences.

You may also consider sending the managing editor of your prefered journal a short note expressing your interest and copy of your CV. They will have the most direct access to injecting you into any relevant manuscript review assignment protocol.