You can do this in AdWords.
It's against Amazon affiliate rules to use your aff links for your own items, don't do it.
You can't track conversions with a pixel, the receipt page is not under your control, and there's currently no method for doing this.
If you also sell direct on your site, some of your site shoppers will leak over and buy on Amazon, and that won't track either (like your direct to Amazon traffic).
Put up a notice on your site (or a buy at Amazon button) that takes them directly to the buy page on Amazon and you'll open a site leak of about 25-30% (assuming the price is the same, the shipping and other factors will get them to by there versus direct).
You can use Amazon as the Display URL in your ad, the ad review process sometimes takes a day longer, but you'll be fine otherwise.
If you have low margins, this whole thing is likely a bad idea, the slice AMZN takes is a big bite.
If you have high margins, the legitimacy you engender via Amzn hook is likely worth the margin ding.
For tracking, keep a spreadsheet of spend versus total sales (units and dollars) where if you sell on both Amzn and Direct, track them both and add them.
Experiment with "Buy on Amazon" signs and buttons.
Do the occasional PPC pause test... pause it, watch sales at Amazon drop, unpause, watch them pop right back up - helps the mind sustain it's belief in the implied attribution if you pile up the empirical evidence.
Your margins determine the outcome here - make sure you track your item margin, ship costs, fees, Amazn slice, Conv Rate, leak rate, marketing spend, CPA, ROAS, etc.
For many people, this is a grand waste of money, shifting the buy spot from one place to another.
For others, the incremental gains are worth while.
One thing is for sure, if you're expecting traffic at Amazon to let you escape the PPC costs you incur to drive your own traffic, this whole experiment will make you cry, a lot.
Amazon is outstanding at closing your nearly closed sales.
It's analogous to a affiliate coupon site, who charges you a 15% commission, and keeps the shopper's email address. In exchange, you gain ~12.63% in incremental volume.
Of course, that's actual incremental, not measured - attribution will confuse you, the reports for Amzn will look far better than actual.
Just like the affiliates coupon site's stats.
But no worries, that'll all likely get lost in the wash, so let's not fret, stew or argue.
Net net, assuming your item margins are fairly normal, 35-55%, you'll be bigger, but your profit level won't move much. 10-15% margins, ouch, you can't afford to share. 60-85% margins, easy decision, slam dunk yepper.
Your mileage will vary.