Apple want 30% of in-app purchases so the Kindle app doesn't support buying directly from the app, you'll need to buy from the Amazon web site.
Once you've signed into your Amazon account in the Kindle app any previously purchased books will appear in the Archived Items section of your library. These can all be downloaded to your device, at no additional cost, by clicking on them.
When you want to buy a new book you will need to go to the Amazon site and, while logged in, go to the page for that book. Right below where you normally click the button to add to your basket there should be a drop-down box that lets you select the device you want to send to. Choose your iPhone and proceed to pay as normal. The next time you open the Kindle app on your iPhone it will sync with Amazon and download the ebook.
