Enable Apps from Unknown Sources in Settings
Open Settings
Go to Security & Privacy
Turn on Apps from Unknown Sources for the Silk browser or Files app
Download the four required Google APKs from a trusted source
Google Account Manager
Google Services Framework
Google Play Services
Google Play Store
Install the APKs in this order
Google Account Manager
Google Services Framework
Google Play Services
Google Play Store
Restart the Fire tablet
Open Google Play Store
Sign in with your Google account
If needed, clear cache and data for Google Play Services and Google Play Store
If needed, update Google Play Services after installation
If the Play Store does not open, restart again and try signing in once more