You build both.

are you serious? You've got a better shot of hitting more home runs then Jose Bautista

Focus the majority of your backlinks to the actual pages you are trying to rank and not just the homepage.

Also ensure that your backlinks are all great quality and the pages that you are getting links from are full of unique content.

Is this actually possible?

Home page links are good, deep links are better. And be sure to maximize your opportunities for internal linking where you control the links and the anchor text.

Amazon doesn't do much product page optimization. You can help your cause with unique reviews/content and re-written product descriptions. Amazon's biggest asset is the authority they get from a zillion pages and their incredible intra-site backlinking by using all those recommended products and "people who bought this bought..."

One way you can create something kind of like this is with related posts or related products on your review pages. Kind of like your own mini-version of "people who like this might be interested in ..."

Yea do both. Don't be intimidated by Amazon. In a way you might find this is easier than you thought because often Amazon is ranking purely because of other factors on its domain, like age and authority. The actual page ranking might not be well optimized at all. What this means is that if you can nail your on-page SEO, and then build quality links you can probably easily outrank that page.