For this Achievement, you need to get to Soggy Island. Buy the guide on Leysure island, then head to the Right when you get off the boat. There is a mountain that says "mine" on it. Once inside, you have to get rid of the rat infestation. Simply grab the log, then move clockwise around to get them all into the storm drain. Once that is done, you can start with the mine.
I highly recommend getting the double jump ability as it saves a bunch of time and some headache. Once your in your mine outfit, you can drop rope with Y, and that will help see where you need to go. You can actually just fall where you need to, and double jump to reset the camera and save time. But it's risky, so be careful. The only thing is that you need to get to the bottom, the diamonds are just worth money later and can be skipped if you'd like. Check the video for help with each level, hope it helps!
2:00 Floor 1
2:44 Floor 2
3:24 Floor 3
4:21 Floor 4
5:40 Floor 5
7:00 Floor 6
8:13 Floor 7
9:58 Floor 8
11:51 Floor 9