This achievement is unlocked through getting every NPC into The Breach, which is the hub of the game.
Here is a list of all the NPCs you need. Please let me know if anything is wrong with this, as I am writing this guide after I already have the achievement.
Ox and Cadence - Will appear in a Jail Cell in the Gungeon Proper (Level 2). A random enemy on the floor drops the key. They run the store in the top left corner of The Breach.
Professor Goopton - Will also appear in a Jail Cell in the Gungeon Proper. Sets up shop just to the right of the Gungeon entrance.
Doug - Will appear in a Jail Cell in either the Gungeon Proper or Black Powder Mine (Level 3). Sets up shop just to the right of the central statue.
Sorceress - Will appear in a Jail Cell once you've beaten the High Dragun at least once. She sets up in the top right of the Breach. You can pay her to 'bless' your run, which makes it so you only have 1 gun that transforms every so often.
Daisuke - Another character that can appear in Jail Cells. He also sets up in the top right room. You can pay him to start a challenge run, which gives modifiers to every room.
Frifle and the Grey Mauser - Another pair that will be in a Jail Cell. They post up in the bottom right of The Breach, and give you tasks to kill certain enemies for Credits or Unlocks.
Gunsling King and Manservantes - Yet another group that will be in a Jail Cell. They sit at the top left of The Breach, though offer no services here.
Ledge Goblin - The last character than can be rescued from a Jail Cell. Once you've rescued her once, you can further ruin her life by locating her on the far right side of The Breach and kicking her helmet back into the Gungeon. Each time you do this, she'll be one floor lower than the previous time. Not sure if you have to rescue her all 4 times for her to count towards this.
Trorc - A random shopkeeper in the Gungeon. Once you buy something from him, he sets up shop to the left of the Tutorial door.
Winchester - Runs a minigame in the Gungeon. Not 100% sure on this one, but I think he appears in the Breach once you've played the minigame once. Sits on the far right side.
Tailor - Will set up on the right side of the Breach near a boarded up door after your first flawless boss victory. Once you've unlocked a shortcut, he'll open this room and post up in there. You can pay him to skip floors once you've opened the shortcuts to them.
Blockner - Sits outside the Tutorial room. He (or rather his ghost) will appear here after you kill him in the Gungeon. To do this, see deathproof4289
mentions that you then have to go talk to Ser Manuel in the tutorial room after fighting Blocker to actually make him appear at the breach.
The Lost Adventurer - Likely the last character many of you will obtain. To unlock him, you have to help him out on every floor of the Gungeon, which is elaborated on in the guide for his achievement here.
Edit: The Advanced Gungeons and Draguns update added Tonic, who is just another NPC who can be found in cells. Once rescued, he'll be in the same room as the Sorceress and Daisuke. Not sure if he's required, but I'd imagine most folks would have him before The Lost Adventurer at any rate.
Edit2: The Farewell to Arms update has added a new NPC named Bowler. Like Tonic, he's found in a cell and will move to the top-right room when rescued.