This method lets you run Google Apps but you may face some hiccups with them so be warned.
First and foremost Enable unknown apps in the settings. To do this drag the notification bar down and choose more. Than tap device. In the device setting set Allow installation of Application to ON.
Download the APK file from good sources which you trust like XDA Developers forum or other such sites from the Kindle Fires Web Browser. Some developers let you download the APK directly from their own website. If the current version of App doesn’t run, try a older one. You'll need to get hold of some extra files besides the APKs themselves, specifically GoogleServicesFramework.apk and the GoogleLoginService.apk, both available at that link.
You'll also need to install ES File Explorer from the Amazon App Store on your Kindle to take care of file transfers and to launch the APKs.
Use ES File Explorer to manage APKs and app installations.
Plug your Kindle Fire HD into your computer via a USB cable to transfer the installation files and get the Google apps up and running. Copy the GoogleServicesFramework.apk and the GoogleLoginService.apk files over to your Kindle Fire HD in the default downloads folder, then use ES File Explorer to launch them. Confirm you'd like to install both apps, then reboot your Kindle Fire HD so that the changes can take effect.
Transfer the APKs to your tablet following the previous instructions, and again use ES File Explorer to launch and install them. Some Google apps will work fine, others may show one or two bugs depending on how they interact with the Android operating system.
Kindly download the APK files from renowned and trusted sites only. XDA Forums is a good and trusted site. Check out the Apps provider’s webpage if they give out the APK, they generally do. Not all Apps may work but something is better than nothing.
Meanwhile Amazon has gone ahead and launched a App for iOS, read about it here. Please Post your comments on this article