(1) look into the air above the ledge you want to climb onto;
(2) raise both your arms by pressing and holding both left trigger and right trigger (your arms reach out in the direction you are looking);
(3) jump up and forward onto the ledge, thereby sticking your arms onto the top of the ledge;
(4) look down toward the ledge, which will cause your arms to move down, thereby pushing your body up and onto the ledge.
I got this achievement within the first hour of normal play (not grinding).
It's pretty awkward, but that's Human Fall Flat for you!