Android: Install Google Play Store on Amazon Fire tablet

Assuming you know how to sideload/install an apk. To get Google Play Store to work, you need to install 4 apps in the following order. No root access is required. The key is to grab the right version for your device.

1. com.google.android.gsf.login - Google Account Manager
   
2. com.google.android.gsf - Google Services Framework
   
3. com.google.android.gms - Google Play Services
   
4. com.android.vending - Google Play Store

For Amazon Fire HD 8 2017 [7th Gen], Fire OS 5.4.0.1, Build 50.5.7.9_user_579278720

1. Google Account Manager, 5.1-1743759
   
2. Google Services Framework, 5.1-1743759
   
3. Google Play Services, 11.3.02 (240-161239932)
   
4. Google Play Store, 8.1.25.S-all [0] [PR] 163906778

For Amazon Fire HD 8 2016 [6th Gen], Fire OS 5.3.3.0, Build 49.5.7.0_user_570208520

1. Google Account Manager, 5.1-1743759
   
2. Google Services Framework, 5.1-1743759
   
3. Google Play Services, 9.8.77 (230-135396225)
   
4. Google Play Store, 7.8.16.P-all [0] [PR] 155590935

For Amazon Fire HDX 8.9 2014 [4th Gen], Fire OS 5.3.3.0, Build 22.5.7.0_user_570208620

1. Google Account Manager, 4.4.4-1227136
   
2. Google Services Framework, 4.4.4-1227136
   
3. Google Play Services, 11.0.55 (030-156917137)
   
4. Google Play Store, 7.8.74.P-all [0] [FP] 157113528