Fallout 4

After entering Vault 88 and rescuing the Overseer, you'll shortly unlock the quest "Explore Vault 88". All you have to do is clear huge piles of debris to reach the areas, find the control cards, and bring them back to the workshops in their associated sectors and you'll be done. There are three you need to find and activate and they are easy to find seeing as they all have waypoint markers. After activating the final sector, you will complete the quest and get the achievement!

Bring stimpaks, radaway and weapons because there are mirelurks, mole rats, radscorpions and ghouls along the way!

So long as you are a Level 20 or higher, the DLC's first quest, "Vault-Tec Calling" will be added to your journal. After completing this short quest, you will soon enough enter the Vault and save Overseer Barstow.

After speaking with her the first time, you will trigger the "Model Citizen" quest, and after turning on the distress beacon during this quest, "Explore Vault 88" will unlock, too. During the latter quest, you will need clear out rubble that's blocking the entryways to each "Sector," North, North-East, and East. Follow your compass and make your way around the underground tunnels of Vault 88 to unlock the three workshop stations in each Sector.

Your compass will guide you in finding each station with ease; however when you try to interact with a workshop station, you'll be tasked with finding the nearby control board in order to power it up. The control boards are marked on your compass. Once all three workshop stations are powered up and available to use, this achievement will unlock and the "Explore Vault 88" quest will be completed.

for this achievement you need to activate all of the work shops around Vault 88. Before you are able to do this you will need to follow the quest line for the DLC until you get a quest called explore the vault. *warning* I would make sure you have about 5-10 stimpaks and rad-aways as there is lots of high leveled creatures and radiation in the vault.

The quest will guide you were to go but if you want to try to avoid the big fights pick up the control board before you go to the work shops. The first one I went to is at the far end of the vault down a train tunnel as you go down it instead of turning left to join another tunnel as your tunnel is blocked go to were the rubble is and there is a room on your right and the control board is there.

The second control board can be found on the way to the next work shop as you head towards it you will come to a set of stone stairs you have to go up under them is a room and in that room you will find the next control board on the shelf on the left.

The last one is found in the train station if you keep heading through the vault past the 2nd work shop, you will find the 3rd work shop first if you go past that you will get to the train station, go to the right and you will find the control board sitting on top of a box next to the pillar.

If you would like a guide on where to find any of the control boards or would like a video guide in general check out my video I have posted below

Simply follow the Explore Vault 88 questline.
Video below shows the locations but it's pretty easy to find.
Make sure to take enough ammo and stimpaks tough Mirelurk Queen in East Sector.