To make sure that you have plenty of fuel enough to kill 20 spiders, be sure to scavenge thoroughly for white plastic jugs and white gas cans that contain fuel. Combining the Flame Thrower and your headlamp seems to be the quickest way of killing them though it does also cause them to run away from you. All 20 spiders must be killed on one playthrough.
Apparently, only the spiders with a round abdomen count toward the achievement. The ones with the spiked tails similar to a scorpion’s do not count, so don’t waste fuel on them. The best strategy is to simply keep gathering fuel and not use the flame thrower at all until the ending sequence of the level when you are waiting for the escape elevator to arrive. You will only be attacked by spiders with round abdomens, and they constantly re-spawn, so there will be no shortage of them to kill. Get 20 kills here before leaving and the achievement is yours. But be careful to stay away from the door of the elevator once it arrives as getting close will trigger the escape animation, ending the level.