How can you download your Amazon video library

Amazon does let you download some shows and movies but it looks like it's aimed at mobile devices like tablets and phones and not the Fire TV. The most straightforward way is to download to a phone/tablet and cast them.

A less straight forward but probably more effective way...If you have a spare PC that's always on with a big hard drive you can use DVR software like PlayOn to record them to your PC (Desktop version, not Cloud) LINK /

You'll have to go through the steps of setting it up and then downloading the videos on your PC. But once that's done you can use the PlayOn app on Fire TV to access the shows and movies you recorded to the PC. PlayOn also lets you download from most services like NetFlix, Hulu and the different cable and network TV channels with supported apps if you have a cable login.

Another thing to consider that bypasses your data altogether is a DVD or Blu Ray ripping program. Get the movies you want on DVD or Blu Ray, rip them to a movie file and access them with a streaming media server. PlayOn can do this but Plex has a free version and it's a much better interface. Like PlayOn this will require you to have an available PC to stream to your FireTV (or whatever streaming box/stick you use).