The below guide will show how to jailbreak the lastest version of Amazon Fire Stick (as of November 2019). This process has been broken down into 3 phases to help simplify the process.
Phase 1 – Installing Downloader App
From the Home screen select the setting option in the top menu.

Within the settings menu select the Device icon ( Or My Fire TV)

In this sub menu scroll till you find ‘Developer Options’

In Developer Options select ‘Apps from Unknown Sources‘

Switch this option on by clicking ‘Turn On’