As of patch 2.4 a key is only needed to open the gate to the Karazhan instance. The masters key can be obtained through a series of quests starting outside the instance. The instance is at the southern most part of deadwind pass at the end of the river. You can begin this quest line at Level 68, but you must be Level 70 to enter Karazhan.
1. Arcane Disturbances (Deadwind Pass)
Speak with Archmage Alrturus outside the entrance to Karazhan, who will give you the quest Arcane Disturbances. Its best to pick up the quest Restless Activity also while your there.
(48, 78) Go down one of the two enterences leading down into the ground. One of these leads to the pond whilst the other leads to the well. When you get to the pond/well use the Violet Scrying Crystal.
(47, 75) When you return to Archmage Alrturus you should have enough ghostly essenses to be able to hand in Restless Activity along with Arcane Disturbances.
Get the follow up quest Contact from dalaran
2. Contact from Dalaran (Alterec Mountains)
From Ironforge fly to southshore in the hillsbrad foothills, go north into the alterec mountains. (15, 54) Speak with Archmage Cedric, and complete the quest. Get the follow up quest Khadgar
3. Khadgar (Shattrath)
(54,44) Speak with Khadgar in the terrace of light
Get the follow up quest Entry into Karazhan
4. Entry into Karazhan (Auchindoun - Shadow Labyrinth)
Form a group to go through the 5 man instance shadow labyrinth in Terokkar Forest. At the end of the instance BEFORE you kill the boss Murmur, go to the left hand side of the boss. Here against the wall, you will find an arcane container. When you open the container an elite mob will spawn, so make sure your group is there to help you. The mob holds the key not the container.
(54,44) Return to Khadgar in the terrace of light, hand in the quest and get the follow up The Second and Third Fragments.
5. The Second and Third Fragments (Coilfang Reservoir and Tempest Keep)
The second fragment: Form a group to go through the 5 man instance steamvaut of coilfang reservoir in zangermarsh. It would be best if the group only needed to do the first boss, or were also on the same quest as you, otherwise you may end up needlessly going through the whole instance. The arcane container can be found passed the first enterance in the water. There is no need to kill the first boss if you dont want to.
The third fragment: Form a group to group to go through the Arcatraz of tempest keep, a 5 man instance in netherstorm. You must have your flying mount to get here. You cannot be Warlock summoned. Either one person in the group must have the arcatraz key, or you need a rogue with maxed out lockpicking to open the door. The arcane container can be found just before the first boss Zereketh the Unbound, to the right as you enter the room.
(54,44) Return to Khadgar in the terrace of light, hand in the quest and get the follow up The Master's Touch.
2nd Key Fragment
3rd Key Fragment
Before you can go to the next step, you must have saved Thrall in the Old Hillsbrad quest line.
6. The Master's Touch (The Caverns of Time - Black Morass)
Form a group to go through the 5 man instance black morass in the caverns of time. Make sure all of your group have done the black morass attunement first. Complete opening the dark portal to speak to Medivh in the instance, he will then give you the masters key.
Get the follow up Return to Khadgar
7. Return to Khadgar (Deadwind Pass)
(47, 75) Speak with Archmage Alrturus and complete the quest.
You are now attuned to Karazhan.