First things first, you'll need to enable installation of apps from unknown sources on your Fire Tablet. This might sound a bit dodgy, but trust me, it's a necessary step to get the Play Store up and running. Just head to your device's Settings, then Security, and toggle the switch to allow apps from unknown sources.
Next, you'll need to download and install four specific APK files: Google Account Manager, Google Services Framework, Google Play Services, and the Google Play Store itself. Don't worry, I won't make you search for them - you can find them all on a reputable APK download site, like APKMirror.