Background - When I had a day job, I convinced the e-commerce store I worked for to list most products on Amazon, too - so I managed roughly 200-300 physical product listings on Amazon at any given time. For a while after quitting, I optimized listings for a number of high-volume Amazon sellers (it got boring fast).
What I noticed:
-You have to optimize your listings for 2 different things - the search engines, and Amazon's internal search engine.
-The 80/20 rule definitely applies. Sometimes you'll spend 5 minutes putting an item on Amazon and see steady sales every day for months or years. Other times, you'll put up a product and be lucky to get one sale/month. Oddly enough, I found only slight correlation between top sellers on the e-commerce website vs. top sellers from the Amazon listings.
-The internal search engine has definitely changed a bit over the years, but some things seemed pretty consistent:
--Keyword in product title
--Keyword in product description
--Free Prime Shipping or not (Prime product seem to be given slight preference in rankings)
--Sales rank
--Star ranking
--Sales in comparison to similar products
--Possibly keyword in reviews, but I could never be 100% sure about that. It definitely helps for external search engines, though, and if you like doing things the shady way, you can always buy a few reviews or give friends/family money to buy the item and place specific keyword-heavy reviews. You can help those stay on the main product page by making them very useful and by soliciting "helpful" votes so Amazon feels they're the best reviews for the product.
--Pricing (in cases where someone else was selling the same item, only the cheapest would be displayed prominently)
--Product tags seem to help Amazon figure out more about your product, but I never really found them to influence internal searches (good for getting your keywords on the page for external search engines, though). I'm not sure if they still have it, but they used to have a function on the seller side where you could suggest search terms for your product. That helped, especially when you couldn't get all the search terms in the title easily.
-Shipping something to Amazon's warehouse and having them do fulfillment is a VERY good idea for anything expensive enough to make the numbers work (they charge a little more when you do it that way). When I was doing a lot of Amazon stuff, there were some products whose sales increased 5-10x just by making that change - and that was at a time before they started giving away Prime memberships to students and adding all the benefits for Kindle/streaming video users. By doing this, you're making it cheaper for customers to buy AND your product gets more exposure (there's even a "Prime eligible" search option).
-On low to moderate volume products, it's relatively easy to manipulate their various cross-selling recommendations. You'll need a few Amazon accounts and some extra cash (you can always refund/re=sell the items later).
1. Frequently bought together (and "Customers Who Bought This Item Also Bought")- pick an item or two that goes nicely with your product. Buy it n the same transaction. It should be a complementary product rather than a direct competitor. As long as you aim for mid-volume or low-volume products, it doesn't take a lot of purchases to manipulate this (usually less than 5, in my experience) and get added exposure on the listings of other products. The cross promotion at the top of the page has been, in my experience, much harder to manipulate than the one with more products that's down lower on the listing (makes sense).
2. What Other Items Do Customers Buy After Viewing This Item? To make sure you show up on this section of other product listings, be sure that each person you enlist to buy your product clicks on the listings of your top competing products first.
