*Note* As of Patch 2.4.0, having [The Master's Key] is no longer necessary for entrance into Karazhan. It is, however, necessary to have the key to be able to open Karazhan's gates.

Step One:
Be at least level 68 and then go talk to Archmage Alturus and pick up the quests Arcane Disturbances and Restless Activity.

Step Two:
After completing the previous quests, Alturus will give you Contact from Dalaran. Take [Alturus's Report] to Archmage Cedric at Dalaran.

Step Three:
Cedric will then send you to Khadgar in Shattrath City.

Step Four:
Khadgar will have you retrieve the three fragments of the apprentice's key (First fragment at the last boss in Shadow Labyrinth. Second fragment in a corner underwater near the first boss in The Steamvault. Third fragment in a corner of the room of Void Walkers in Arcatraz.).

Step Five:
Complete Caverns of Time: Old Hillsbrad and rescue Thrall. (This can be done before the other steps.)

Step Six:
Complete Caverns of Time: Black Morass and protect Medivh. After the last boss fight, talk to Medivh and have him restore your apprentice's key. Take the [Restored Apprentice's Key] back to Khadgar to receive [The Master's Key].