This is a guide for getting LineageOS (an older and unofficial version) onto an Amazon Fire 2015 7" tablet (also known as the 5th generation 7" Amazon Fire or under the internal codename "ford").
NOTE: This guide was originally written before someone figured out how to unlock the bootloader. There is now a newer method. Here it is very briefly. The old guide can be found below.
First unlock the bootloader using this guide.
Once you're booted into TWRP download the latest LineageOS 14 release for your device here.