Aliens: Fireteam Elite

The easiest way to get this achievement quickly is to either play Horde mode as a level 3+ Technician with a V99 Incinerator or run through 4-3 as a Demolisher with the Volcan flamethrower(eggs/facehuggers count). Horde unlocks after you complete all 12 Campaign missions and as long as you played Technician on a couple levels with maybe an exp challenge card you will have the flame turret unlocked by then. If you want to level up the V99 and Volcan I recommend you do so on Standard difficulty because they're basically a no-no on Intense+ as they run a very high risk of friendly fire and can also obstruct the vision of your allies.

Fire up a private game of Horde mode on Standard difficulty and bring a decent pistol(Kramer or Type 95) and the V99 with fire rate attachments. If you bring a friend both of you can get progress and 2 flame turrets will give you a higher success chance of completing the 10 waves for additional rewards, otherwise solo sometimes gets interesting on the elites. The reason I recommend Standard is because on casual the bots have better aim and enemy HP is less so they'll steal your turret's kills, thereby taking longer. Equip perks that increase the damage of your turret(Force Multiplier) and perks that increase the duration/reduce cooldown of the coils. Put the flame turret on the big green box facing away from the ammo crate and keep a coil next to it based on the direction they're coming from. Once you get to wave 5 or so things could get tricky with Crushers so try to keep a coil or two in reserve. The consumables Hardened incendiary turrets and rounds do not help progress.

It doesn't seem to track kills if you are just hip firing with the heavy flamer. You may notice you don't get hit markers when shooting this way. Make sure you hold LT while shooting and you do get hit markers and the kills do count. The tracker updates in real time so you can check if they are counting. Now go purge those xenos in the name of the emperor! Wait, wrong franchise. Oh well, close enough.

In this video guide I will show you how to kill 1000 enemies with fire. First thing you will need is to go buy the Type 99 Incinerator. Next use the technician class and change the turret type to incinerator. If you don't have it unlocked you will need to level up the technician. Now go to The Gift of Fire chapter and select mission 3:Boarding. Now kill everything you can with the flamethrower and turret. Should only take a few attempts on this mission and the achievement will unlock.