This is how you install the Google Play Store and Google Services on the Amazon Fire Tablet.
On the tablet go to Settings -> Security -> Apps from unknown Sources (Turn On)-> Press OK.
Open the browser on the tablet and go to these two links:
Step 1 – Click this link http://bit.ly/1TzWVzq and download and install the 4 files in order.
Step 2 – Click this link http://bit.ly/1X0RtdZ and download and install the 2 files.
Step 3 – Reboot Tablet
You can now install anything from the Play Store.