The following was done on an Amazon Fire HD 8 (6th Generation) but should also work on any Amazon Fire tablet released from 2014 onwards (4th Generation and above) that's running Fire OS 5.
NOTE (1): To install the required APKs, you'll need a file manager, so get one from the Amazon Appstore, I recommend Total Commander - https://www.amazon.co.uk/dp/B00H4IHJ8M/
(TIP: Don't use or install ES File Explorer as it can interfere with downloads and installing APKs.)
NOTE (2): You'll also need to enable 'Apps from Unknown Sources' found under Settings > Security of the Fire OS.
(TIP: Some users recommend rebooting after enabling this option, but I didn't and had no issues.)
Now all you need to do is download the 4 APK files below using Amazon's Silk browser and install them in that same order, then reboot your tablet and afterwards open up the Google Play Store and log in with your Google account:
1. Google Account Manager 5.1-1743759
2. Google Services Framework 5.1-1743759
3. Google Play Services 10.0.84 (230-137749526)
4. Google Play Store 7.5.08.M-all  [PR] 146162341
Now you should have the Google Play Store up and running.