How to approach a Book Review Editor
Yes, it's entirely appropriate to volunteer to serve as a reviewer.  In doing so, you should be clear about your current background and current position (e.g. that you have a PhD and are working as a postdoc) and your specific areas of expertise.  The worst that might happen is that you'll get no response or be told that they want more experienced reviewers.  

Book review columns vary tremendously in the number of books that get reviewed and the depth and quality of the reviews.  In some cases the reviews are brief (less than 200 words) and descriptive, while in other cases the reviews are longer and include more of an evaluation of the work. Some book review columns solicit in depth reviews from experts in the field, while other columns (particularly ones that review a wide variety of books including textbooks) may use reviewers who aren't necessarily experts on the topic.  

In my opinion, it's best to try to limit yourself to reviewing books in areas where you really are an expert- I've been asked to stretch further than I'm comfortable with in a few cases, and I didn't like the results.