Baldur's Gate II: Enhanced Edition

Complete the quest "Inside the Planar Sphere" with a Main Character whose class is Mage or Sorcerer.The stronghold is unlocked by accepting it from Lavok at the end of the quest.

To get the Wizard stronghold, choose Mage as the class for your main character. Go to the Government district in Athkatla and speak with Madeen, whom is standing in front of the Council of Six Building. Madeen tells you to go inside and talk to Tolgerias, whom in turn will give you the quest "The Hunt for Valygar Corthala". Go to the Docks district to find clues in Valygar's home, then travel to Umar Hills. Find Valygar's cabin to the northwest. Either recruit or kill Valygar, then travel back to the Slums district in Athkatla and open the door to the Planar Sphere. Once inside, you'll receive the quest "Inside the Planar Sphere". Complete the quest, ending with killing Lavok, and on the way out the Cowled Wizards will offer you to take control of the Sphere. Accept, and the achievement will unlock.

Note: This can be done immediately after the prologue and speaking with Gaelan Bayle.