The Fire TV 1 running software versions 220.127.116.11 or newer cannot be rooted.
There is a hardware rooting method for advanced users that involves disassembling the device and soldering a board to the Fire TV’s memory in order to root by writing to the device’s eMMC chip directly. I do not have a guide for this method because it is a delicate procedure and not recommended for most users. Most users wishing to root should block updates and wait for a new rooting method to be discovered. Staying on older software versions increases the possibility for your Fire TV to be rootable in the future. There is no way to know if or when your Fire TV will be rootable.
The good news is you do not need to be rooted to sideload apps like Kodi. Sideloading does NOT require rooting.