This achievement is obtained on the second half of the mission 'Payback'.
Near the end of the level, after you land your helicopter, you will come up on a cave. Woods should say something like 'Pick up that flamethrower.', and there should be one on one of the dead guys at the entrance to the cave. It will be an AK-47 with a flamethrower attachment.
Just switch to the flamethrower by pressing left on the D-pad and kill 10 guys in the cave!