Can I see my purchase history from my iPhone

Not your full purchase history, that can only be done via a computer's iTunes - on your phone you can view the last 90 days via (though it doesn't appear to have updated recently, mine doesn't show my last app download and other people have also posted about it) http://reportaproblem.apple.com

Yes for the last 90 days

Click here: Report a Problem

Otherwise you'll have to use your computer iTunes.

Please see this link: Download your past purchases - Apple Support