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 No Laughing Matter.
There will be three quests you'll need to complete for the achievement. You will need a minimum of 40+ intellect to complete the achievement.
First, The NPC will task you with taking a letter they give you and delivering it to a random inmate. Head back to this same NPC. 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 obtaining a bow tie. Go to the marked on map desk and obtain it then deliver it to them.
Third, They will task you with crafting flimsy cutters. Go to the 3 desks marked on map to obtain 2 files and a duct tape. Craft the cutters and return them to quest giver NPC and your achievement will unlock.