Once you've completed Grey Mothers quest line you should have a climbing rope and a map marker showing where to go next. Before you leave Milton make sure you get the achievements for opening all of the safety deposit boxes and for completing Lily's story.
Also beware of the side mission for filling Grey Mothers fridge. Anything you put in the fridge is lost to you. Don't make the mistake I did and leave yourself continuing the story with an empty belly!
When you're ready to leave and have prepared your pack (inventory space is quite tight but try and pack a lantern) head to the map marker for the climbing spot and you'll find a big rock where you can attack the rope and get down the cliff face. Obviously you rope snaps so there's no going back.
You'll then meet Methuselah again (the old man from the gas station) so have a chat with him then head east / southeast and you'll find another rope hanging down a cliff face. Climb up this rope, walk across a fallen tree that bridges a crevasse and head into the cave. Get out your lantern or torch and follow the linear route through the cave. Once you exit at the other end the achievement should unlock.