The villager on the left of the door of the first house you see when walking out of the starting area is the guy that has this quest, you don't have to talk to him to start it. There are 3 houses in the village with spiders in that you have to clear out.
I recommended starting with the warrior since he is the only one that starts with a weapon, unless you buy one with points from a previous playthrough. Equip your weapon and go into the first house in front of you and kill the one spider(attack with 'B'), check the chests and you might find a better weapon.
Head back outside and turn right, the house behind the villager in front of you is the next one. Head inside and kill the one spider then head back outside. Turn right and the house you see on your right is the third house. Head inside and destroy the web and then kill the 2 spiders.
I believe the achievement unlocks as soon as you kill the last spider, if not, go talk to the village I pointed out in the beginning of this guide. Which is to the right if you walk out of the third house and head straight. He will also now heal you whenever you talk to him. Let me know if I can improve this guide in the comments.
