Currently dropbox itself has to be sideloaded first (apparently they are still waiting on approval to get into the amazon marketplace). Once it's installed you can push your backed up apk file by browsing to it in a file explorer (I use astro), long press file > send > dropbox.
A minute later it will be on your fire, in the dropbox folder, ready to install.
Before this I was emailing the files to myself.
A minute later it will be on your fire, in the dropbox folder, ready to install.
Before this I was emailing the files to myself.