- Obtain and enter any of the following inn rooms.
Permanent Tenant is awarded as part of the Room to Spare quest, which rewards you with a free room at an Inn, and also serves as a tutorial to Player Housing in general. Since you need to complete this quest in order to purchase any other homes (including the two Inn rooms in the other Alliances), you really only have one choice of room to acquire in order to get this achievement the first time: Mara's Kiss Inn Room The Rosy Lion The Ebony Flask Inn Room.
After acquiring the room in your home Alliance, you can purchase the other two for 3,000 each, or repeat the quest on other characters in the other Alliances to earn the others for free as well. These rooms cannot be purchased from the Crown Store. Since all homes are shared between all characters in the same account, your other characters after the first need only enter one of these rooms in order to earn the achievement.
Note that completing this quest in Vivec City and acquiring Saint Delyn Penthouse or Alinor for Golden Gryphon Garret will not earn you this achievement, but you can still easily get it if you have acquired one of the other apartments on a different character or paid the gold for it yourself, just by entering the room.