This video shows how to easily install the Google Play Store onto a Fire Tablet in a matter of minutes. This is the easiest and fastest way to do it. Everything can be done from the Fire itself, no computer or rooting required.
Install Directions: http://blog.the-ebook-reader.com/2016/07/02/how-to-install-google-play-on-fire-tablets-the-super-easy-way-no-pc-or-root/
Update: If your Fire tablet has Fire OS 22.214.171.124 or 5.3.2 or if you are suddenly experiencing problems after updating, then use these newer apps from this post at the XDA forums: http://forum.xda-developers.com/amazon-fire/general/how-to-install-google-play-store-fire-t3486603
Amazon affiliate link (helps fund videos): http://amzn.to/29drBfF
Download apk files (old link): https://archive.org/download/GooglePlayServices4.4.521174655036