This quest will start immediately after stumbling into the vault in the basement of Malden Middle School. Kill the guards outside then take the elevator down to reach the vault itself. Your objective is to fully explore the area and reach the Overseer's terminal.
You can't access the security room without the Lab Access Card. Make your way to the atrium where more gunner forces await.
The reactor room to the south doesn't have anything, aside from a broken walkway leading to the Overseer's room. The upper level has only residential units with no notable loot. Make your way to the north, to the lockers. Open the sliding door there then follow the path until you reach the connecting room to the lavatory and observation room. More gunners will be waiting inside.
Head to the Observation area and you'll find a diner with the Gunner Commander inside. Kill the commander and get the Lab Access Card from him. The terminal near the entrance of this area can activate a Combat simulator where you can shoot targets at the diner.
Now backtrack to the security door and unlock the door using the access card. The Overseer's room to the right requires an Admin Access Card so go through the sliding door to the north then continue to the observation hall with Glass walls and flooring.
Kill the gunners in this hall then continue to the last room to the west then kill the gunners inside to loot the Vault 75 Admin Access Card. Here, you can also find the Science Bobblehead.
Backtrack to the door requiring the Admin Access Card and use it there. Continue forth and kill the Gunner Corporal. From there, you can access the Overseer's room to the south and an Advanced door to the west. Head to the Overseer's room and kill the last Gunners there, including a Commander. In the Overseer's desk, there's a Master lock and to the north of the room, there's an Expert sliding door. You can unlock both using the Overseer's terminal. Also, upon accessing the terminal, the quest will be complete.
Inside the Expert-locked door is a green steamer trunk, a Master-locked floor safe under the bed, and an issue of Grognak the Barbarian. The Overseer Safe Key is also found in the same room, between the red toolbox and the locker. After looting the items, you can now leave the area. Upon fully exploring the area, the map marker for Malden Middle School will now be replaced by the map marker for Vault 75.