Far Cry 3 Walkthrough -Kill 50 Enemies with Flamethrower
The flamethrower can be purchased from the store fairly early on in the game. You also receive the weapon just by playing through the main storyline as well. So when you have it in your possession, equip it, and then kill enemies with it.
You have to kill a total of 50 enemies with the flamethrower, which can take quite a long time.