If it's a free app on Google Play you can use one of the
APK download sites by pasting the GP URL for the app in and downloading the APK file, then sideloading it to the Fire. Not every sideloaded app works or works well (like Vudu) on Amazon's custom Android fork, but most work just fine.
As to the Google Play store on the Fire here's a method, although I've never tried it, I've just sideloaded the few apps I wanted...
http://forum.xda-developers.com/fire...6-hd7-t2933523