First, Act Like a Book Reviewer: Review Books, a Lot!
Being a paid book reviewer sounds like a plum job for writers, who generally love reading as much as writing. It's not a pipe dream. There is paid work available for book reviewers.
The first step is to obtain books on your own (at your own expense) and publish reviews on open platforms like Goodreads or Amazon.
This helps the writer in several ways. It keeps you on top of the recent releases in your interest areas and genres. This is important because reviewing focuses on recent releases (with a few exceptions). It also teaches you the process of writing a book review. Interacting with other writers, reviewers and readers will help to shape your prose. You'll also get the chance to observe their review styles.
You may also begin to develop a following of fans who appreciate your reviews and writing style. It is from this following that you build your audience for later endeavors.
Develop Your Own Book Review Outlet
Once you've got the hang of reviewing books, you'll want to develop a site or niche where you can publish your work yourself, such as a fan page or a blog. It helps to establish you as an expert, and puts the focus on you as a reviewer/brand, as opposed to Amazon reviews, which people may not associate as much with the review authors. It also serves to gather your prose/writing in one area/profile, which you can then use later as evidence of your skills.
Starting a book review blog puts you in full control, and you may begin earning pay for your reviews that much earlier. Also, you can then open your blog/site to authors who are seeking reviews or doing blog tours. It might mean you'll then start getting your books for free from the author/publishers. It also means you'll be privy to new releases—previews that may not be available to the public yet. Again, this will serve to build your expert credentials. Another bonus is that you begin to build relationships with those authors/publishers.
Often, the books you receive from these relationships are Advance Reader Copies. It is a "rough draft" of the book produced for first readers and reviewers. These cost less to produce and can be sent out early, even if the final book isn't completely done.
If you can cultivate a relationship with a publisher at this point, you may have the good fortune of being put on their marketing/publicity list. It means they'll send you emails or catalogs asking you which of their new releases you'd like to receive.
Gather Your Documents Together
It is now almost time to start chasing paid opportunities. Gather together your best reviews—the ones in which your prose just flows, and your passion is evident. Format them attractively and save them as a PDF. Also, if this particular review is on a website, save the URL, too, as some book reviewing jobs ask for links.
Next, prepare a resume focusing first on your book reviewing credits and skills, and second on your other writing credits and skills. Some outlets that are in a position to pay book reviewers may request a traditional resume. However, they're not looking for a list of every job you've had in the past 10 years. They want evidence of your writing/reviewing ability.
Your last document will be a cover letter. It will generally be the text of an email responding to open jobs/projects. Put together a basic cover letter for a generic book review position, and then customize it for each potential project.
Pursue Paid Book Reviewing Jobs/Projects
Even if you're hoping to focus on reviewing for specific magazines, or reviewing for a particular leader in the industry, build up your credentials by getting some paid work.
Look for specific freelance writing jobs that ask for book reviewers. If you find an opportunity you want to take up, respond specifically (tailor your cover letter) and quickly (as these guys get inundated fast). Don't be discouraged if you don't get a response. It is a volume game.
You may pick up some regular clients in this manner. Generally, these companies, sites, or publications have a relationship with the publisher, so you may receive the review copy from your new client.
One side note here. Some authors and companies pay for positive reviews. It is an ethical consideration for you if you want to continue and be accepted in the book reviewing field. Generally, a professional reviewer is expected to be an impartial source.
Pitch to Magazines, Journals, and Newspapers
Now that you're an established book reviewer with a few paid clips in your portfolio, the next level is getting your reviews placed in publications—both print and online. It might net you a wider audience, and certain publications gain you some credibility as a writer/reviewer. Also, print publications may pay a bit better. Aim for the big guys: Booklist from the American Library Association, Kirkus Reviews, and Publishers Weekly. You can start with a regional publication and build up.
Next, you want to query magazines. There might be a bit of variety. For example, some editors may want to see the review in total, as opposed to pitch or query letters. Some may list you as an ongoing potential reviewer, and send books that match your stated interests or expertise areas. Some may come to you with potential titles, whereas some may let you pitch titles that you think their readership would like.
Finding outlets that accept book reviews is similar to finding magazines to publish your other written work. Start with the "Writer's Market" or visit the magazine's website.
Keep Current and Get Educated
Retain your status as a paid, professional book reviewer by staying current and relevant in the field. Keep on top of new releases, specifically those in your favorite genres. In addition, most major book sites like Amazon and Barnes & Noble have sections regarding upcoming releases. Follow publishers on Twitter, or sign up for their marketing emails to get insider scoops.
Consider joining the National Book Critics Circle, a professional association for book reviewers. They offer education and networking resources for reviewers, along with updated listings of potential outlets. The NBCC is open to professional reviewers who can show published review clips. It's a club you want to be in.