Google Services on the Amazon Fire HD 8

xda-developers has a post, Installing Google Framework/Playstore without Root (5th Gen Amazon Fire 2015), that includes a couple different sets of instructions.

What worked for me was grabbing the zip download and then:

  1. Make sure you're a developer and you allow apps from unknown sources.
  2. Install ES File Explorer via the Amazon Appstore. It's something I install on all my Android devices.
  3. Extract the contents of the zip you downloaded from Mega somewhere.
  4. Connect your Fire tablet to your computer via a USB cable.
  5. Copy all the files to the Download directory on your tablet.
  6. In ES File Explorer navigate to the Download directory.
  7. Install*.apk
  8. Install*.apk
  9. Install GoogleLoginService.apk
  10. Install GoogleServicesFramework.apk
  11. Turn off and then turn back on the device.
  12. You can try opening Google Play. If it automatically closes (it did for me), install*.apk again. When I did so it asked me if I wanted to install the update, and referenced one of the other services.
  13. You should be able to run the Google Play application now.
  14. When prompted, you can safely update the Google services.

With Google Play installed I was able to login to Google, as well as install YouTube and Chrome to my Fire HD 8.