Disclaimer
Remove the microSD card from your Fire tablet before proceeding as it can cause issues during the installation
Enable Unknown Sources
- Login into the Fire tablet
- Launch Settings
- Navigate to Security & Privacy > Apps from Unknown Sources
- Select Silk Browser > Enable the Allow from this source option
- Close Settings and navigate back to the launch screen
Download and Install APKs
- Launch Silk Browser > Navigate to https://bit.ly/apfiretablet | Alternate
- Download the following APKs for the target Fire tablet
- Google Account Manager
- Google Services Framework
- Google Play Services
- Google Play Store
- Close Silk Browser and navigate back to the launch screen
- Launch the Files application > Navigate to the Downloads folder
- Select and install the downloaded com.google.gsf.login apk > Tap Done
- Select and install the downloaded com.google.android.gsf apk > Tap Done
- Select and install the downloaded com.google.android.gms apk > Tap Done
- Select and install the downloaded com.android.vending apk > Tap Done
- Once all four APKs have been installed, restart the Fire tablet by holding the power button and selecting Restart
- After restarting, log back into the tablet and launch the Play Store app
- Login with a Google account
Source: https://www.androidpolice.com/install-play-store-amazon-fire-tablet/