You will automatically equip the flamethrower at the start of the level DARK STAR. You will be attacked by Nosalises. Your best bet is to go ham on them using your new toy. You can keep the flamethrower and should be able to obtain the achievement in the next level or so.
However ammo for the flamethrower is pretty scarce so an alternative method is to just reload checkpoint when the Anomaly(blue electric spark bulb thing) appears and again go ham on the monsters. Checkpoint reload does not reset your progress towards this achievement.