The Amazon Fire 5th Gen is finally rooted! The root method is actually pretty simple. All you have to do is download the zip file to a Windows PC/ or Linux, unzip the file, click install.bat and follow the onscreen prompts. Once you are rooted you can install the Play Store which gives you access to all the apps and media you have already purchased from the play store. There are plenty of other things you can achieve with root on this device. There is a full install video at the link below along with all the files you need.
via XDA