This doesn’t even require rooting your Fire Tablet.
On Linux
Enable ADB.
Enable Dev Options on the fire Tab by clicking on 7 times.Locate the “Serial Number” field on this page and tap it repeatedly. Tap it seven or more times and you’ll see a “Developer Options” option appear below it. Tap “Developer Options.”
Get the 1 file from http://rootjunkysdl.com/?device=Amazon%20Fire%205th%20gen&tag=823814-20 called "Amazon-Fire-5th-Gen-Install-Play-Store.zip" , unzip and collect only the files with the apk extensions.
USB or IP connect the FireTab.
Now install and change properties in these orders :
adb install com.google.android.gms-6.6.03-6603036-minAPI9.apk
adb install GoogleLoginService.apk
adb install GoogleServicesFramework.apk
adb shell pm grant com.google.android.gms android.permission.INTERACT_ACROSS_USERS
adb install com.android.vending-5.9.12-80391200-minAPI9.apk
adb shell pm hide com.amazon.kindle.kso
Now you should be able to fireup Google Play store after logging in with a gmail account.
All these are availabel for WINODWS at
http://www.howtogeek.com/232726/how-to-install-the-google-play-store-on-your-amazon-fire-tablet/
I had just read the bat files to skim out the commands for Linux.
