Will unlock this on your way to the following:
The good news is that it's cumulative across all modes so every bit of progress helps. I personally took a pray and spray approach to this. I would hold down the trigger and spray as soon as I picked up the ammo.
Additionally, you can pick up the flamethrower at the beginning of each mission when you leave the base. I recommend doing that so you work on kills right away when each mission starts.