Although the Kindle Fire runs on the Amazon Appstore, you can still sideload quite a lot of Android apps (those apps that are available in the Google Play store), and it requires no rooting. You can side-load apps in all Kindle Fire models, including the latest Fire HD and Kindle Fire HDX models. Below, you can find a complete discussion on how to do it.
Complete discussion on Kindle Fire and sideloading apps:
The Amazon Appstore does have over 300K apps, and it keeps growing. Most of the apps that are popular in other App stores (like the Google Play Store and the Apple App Store) can also be found there. Apps such as Angry Birds, Minion Rush, Netflix, Pandora, Glow Hockey, Subway Surfers, Temple Run, Where’s My Water, Talking Tom, Solitaire, Twitter, Minecraft PE, Candy Crush Saga, etc., can all be found in the Amazon Appstore. You can always do a search to see if your favorite apps are there (you can get the apps to your Amazon account using your computer, and it’ll be on your Kindle tablet).
If you can’t find some apps in the Appstore of Amazon, you can always try to find and side-load the app somewhere else. It’s easy, and no rooting is required; you just have to allow apps from other places in the settings of the device.
Probably the app that most people looked for is the Adobe flash player, since it no longer comes with any new tablet. You can read a discussion here on how to get Adobe Flash on the Kindle Fire HDX.
Finally, I recommend you to read this discussion on Kindle Fire + Google Play store: