You will have to play trough the game at least twice to obtain all achievements during which you have a very high chance of unlocking this achievement as well.
Near the end of the game, on the level DUNGEON, you will be separated from your squad about halfway trough the level. This is the only place where you will encounter spiders.
Now spiders can't be killed by normal bullets. You have to instead shine your flashlight on them which they will take damage from. Make sure to charge your battery so your light does more damage. It takes about 5 seconds of exposure for a spider to start dying. If it becomes red than you are doing the right thing. Keep shining until it stops moving.
You can kill about 5-7 spiders in this area so reload checkpoint when you are done as reloading checkpoint does not reset your progress towards this achievement.