Check out their site:
GO Launcher | GO Launcher | GO
The site is in the form of a blog, just look for the latest release message (I do non-beta) and they will have a link to the apk.
The following is from memory, so you may have to do it a little differently, but this should get you close...
Just enable loading of untrusted content (or whatever they call it) on your Fire, then go to the site on the Fire browser and download the APK. Then click on the APK in your download folder, and you should be good to go.
When you run it, it will ask you if you want it to be the default. I didn't, because for some things I prefer the carousel, so I just click on the appropriate launcher each time I press home, or power-up.