1x Guard Uniform
1x Toothpaste (for Putty)
1x Talcum Powder (for Putty)
1x Lighter (for Melted Plastic)
1x Plastic Comb/Spoon (for Melted Plastic)
1x Fake Red Key (use Red Key + Putty to make Red Key Mold and then combine it with the Melted Plastic)
Find the inmate that's giving out the "My Zombie" quest line. This is an 8 part quest. You need to do all parts. Failing any of the quests will reset your progress to the beginning of the quest line, so if you mess up dashboard before the game auto-saves. Conversely, if the guards wind up confiscating one of the items you need you might as well intentionally fail it by flushing one of the ingredients and start over (you at least keep your Intellect/Strength so it's not a total loss).
For the most part the quests are all fetch quests with some assembly required. You will be assembling quite a few contraband items using contraband ingredients. Keep in mind where the metal detectors are (e.g., kitchen) and don't go through them with any of the ingredients. Also make sure you accept the quest outside of shakedowns because you don't want the guards to take an ingredient you need into the high security contraband room. Specifically, do not accept a quest during roll call or right afterwards. Here's the schedule for the prison if you want to plan around it:
While you're making progress in quests keep on building up your intelligence (North West corner of main building, 2nd floor) as you will need it to craft a bunch of the quest items. Don't neglect your strength and speed either as you will need to beat up guards for the red key.
You will need the red key because one of the fetch quests will always spawn the item in the guard room. The guard room is in the basement and you will need a red key and a guard uniform to enter it. Beat up the guards to figure out which one has the red key, and make a mold of it. In my experience the time to loot a guard is super short since there are medics on the same floor (and don't attempt to beat up guards outside because there are snipers). You will probably only be able to grab the red key and uniform before the medics take the guard away (i.e., you won't be able to put the key back). Make the mold, drop the red key on the floor, and quickly run to hide the mold in your hidden desk compartment. The prison will probably go into lockdown but if you don't have the key on you, you shouldn't get in trouble.
Once you complete all 8 parts of the quest line the inmate will give you a special key (note that it is a contraband item). During one of the free times go to the North West building to the room with a large coffin in it. Equip the keep and press A on the coffin to escape.