Quests in the game are akin to favors. Fellow NPC Inmates with GREEN EXCLAMATION points / marks above their head have quests to give. The quest they have is random. If you are solely going for the achievement you can keep declining the quest they have and then cycle for a new quest until you get one that reads It Belongs in a Museum.
There will be three quests you'll need to complete for the achievement. You will need 70 Intellect to complete the achievement. Intellect can be increased by going to the library and completing a mini-game involving holding the
for periods of time. You can find the Library on the lower left on the map. First
, The NPC will task you with delivering a letter to a random guard NPC. The item will appear in your inventory as soon as you start the quest or if your inventory is full when starting it will drop on the ground at the NPC location. This guard will appear on your map by the sorry game piece symbol (pin). Deliver the letter to the correct guard and then head back to the same quest giver NPC you started with. There can be multiple occurrences of this quest going on simultaneously (i.e. two inmates have this quest) so make sure to go back to the same guy you completed round 1 with. Second
, They will then task you with retrieving and destroying the toothbrush shiv. This can either be a desk or a person. Go to the marked on map desk and obtain the item or beat up the inmate carrying the item (whichever random one it gives you). Take the shiv and flush it down any toilet and you will have completed the second part of the quest. Head back to the same NPC that you've been dealing with and start the third part.Third
, They will task you with retrieving 3 items from random "easy" inmate desks. The Broom Handle, the Molten Metal, and the Iron Bar. Grab these 3 items from the 3 marked desks. Once you have all in your inventory at once it will give you the option to craft a "Makeshift Halbred". You will need 70 Intellect to do this crafting. Craft the Halbred. Once you've crafted it, the quest will then tell you to beat up a random inmate NPC with the Makeshift Halbred. Beat up this NPC and once you've done so, your achievement will unlock! You will not need to return to the NPC for this third quest.