As always in an Android device, you can allow the “Installation of Applications from Unknown Sources” in setting menu and voila, you just need to transfer the APK to your Kindle Fire and navigate to the APK using Estrong File Explorer or any of them (which are available in Amazon Appstore) and you can easily install the app.
You can also use Getjar Appstore, but according to the user experience so far, it is not turning out to be a viable alternative but still you can try and get one of the free gold apps. Just navigate to their mobile page and install their appstore client and you are good to go.
PS: B&N Nook App works great on Kindle Fire (ironic :P)
