How to Install Dropbox on the Kindle fire

Click the gear icon in the top-right corner of the Kindle Fire home screen to activate the Settings window
Click More
Scroll down to Device
Scroll down to ‘Allow Installation of Applications from Unknown Sources’ and move the slider from ‘Off’ to ‘On
You’ll be prompted to confirm you mean to do this; you agree
Switch to the Kindle web browser and go to https://www.dropbox.com/android (NOTE: the ‘https‘ is important!), and click Download App. This will download the Dropbox .apk file
When the download is finished you’ll see a notification in the top left corner. Click it and scroll down to the notification that the download of the Dropbox .apk file is complete. Click the file.
You will be prompted to install the application. Click the install button.
Once installation is completed, start the Dropbox application and log in.