The Amazon Kindle Fire is quite a popular device, but its app selection is initially limited to the Amazon Appstore, which doesn't have every app you'd find elsewhere. If you have a second Android device, you can actually take apps from one device and send them to another without even needing a computer. You will need to have a cloud storage account through something like Dropbox to make this work.
On both the Fire and the other Android device, you'll need to install the ES File Explorer File Manager app. It's free and available in both the Amazon Appstore and Google Play Store. On your non-Fire device, you'll need to install another app called Apk Extractor. This app can repackage installed apps on your device so you can bring them over to your Kindle.
In this example, we wanted to ship over the Logitech Harmony remote app that is available on Google Play, but not the Amazon Appstore. Open up Apk Extractor and find the app you want to export. A simple tap will package up the app you want to grab and send it to a directory on your phone; in our case the APK wound up in /storage/emulated/0/ApkExtractor.
Then launch ES File Explorer File Manager on your Android device and connect your cloud storage account. We used Dropbox in our example, but ES File Explorer supports a number of alternatives, including Box and OneDrive. Click settings, then tap Cloud from the menu, then hit the plus icon on the bottom to connect your cloud storage account to ES File Explorer File Manager.
Now, navigate back to where your exported app lives and and long press it until you see new options for cut, copy, and paste on the bottom of the app. Select copy, then navigate to your cloud storage and paste the app there. You're done messing around with this Android device; now get your Kindle Fire.
On your Kindle Fire, go to Settings by swiping down from the top of the device, then click Applications. Find the "Apps from Unknown Sources" option and hit the toggle button to allow installation of apps from outside the Amazon Appstore. This will allow you to install the APK you exported.
Open up ES File Explorer and connect it to your cloud storage account the same way you did on your other device. Navigate to the Harmony APK and open the file. Your Kindle will ask you if you want to install this application, so click Install. Congratulations, you just sideloaded an app on your Kindle Fire without using a computer.