This should work on any Amazon Fire tablet released from 2014 onwards (4th Generation and above) that's running Fire OS 5.X.
NOTE (1): To install the required APKs, you may need a file manager from the Amazon Appstore, I used TOTAL COMMANDER - https://www.amazon.co.uk/dp/B00H4IHJ8M/
With later versions of Fire OS or SILK, you can just tap the download notifications as and when you download the APKs or use SILKs download manager, removing the need for the above file manager.
NOTE (2): You'll also need to enable 'Apps from Unknown Sources' found under Settings > Security of the Fire OS.
NOTE (3): As of Fire OS 184.108.40.206, when sideloading APKs you'll need to press the POWER BUTTON to lock the device when on the APK install screen and then unlock it again to allow the INSTALL BUTTON to be pressed. (Thanks to whoever found that workaround).
1. Google Account Manager 5.1-1743759
2. Google Services Framework 5.1-1743759
3. Google Play Services 11.5.09 (230-164803921)
NOTE (4): If you have a Fire HD 8 2017 (7th Generation) use this APK below instead of the one above.
3. Google Play Services 11.5.09 (240-164803921)
In future, the Google Play Store and Google Play Services apps will auto update themselves in the background.