To unlock Trash King's vault, you'll have to first go to the level Bio Lab. Use the snake tail in the hole to move the magnet in a specific series of inputs, and you should see a little cutscene of the vault opening in the other room.



The series of inputs is: Up, Left, Down, Left, and then the GRAB button. Immediately after putting this in you should get the cutscene. It may take a couple tries, and the snake bouncing around too much and accidentally hitting buttons more than once, even though it won't move the magnet more than once, may mean you need to restart.

After you do it successfully, finish the rest of the Bio Lab level and continue to the next level (the hallway area now filled with rabbits). Head up from the hallway, to the screen with the raccoon statue and the rabbit with the blue keycard. Before going back to use the blue keycard to enter the Anthropology area, head another screen to the right and enter the now-open vault door in the security area. The achievement should pop as soon as you enter.