You have to be level 70 before you can get attuned with Karazhan. There is a specific order that you need to do the instances in to get attuned, and I'll take you through them step-by-step in this guide.

Here is the instances listed in the order that you need to do them:

1. Mana tombs (you need to loot the key in Sethekk Halls)
2. Sethekk Halls (loot the key to Shadowlabs)
3. Shadow Labyrinth (get the 1st key fragment)
4. Steamvaults (get the 2nd key fragment)
5. Botanica (get the 1st key half for arcatraz attunement)
6. Mechanar (get the 2nd key half for the arcatraz attunement)
7. Arcatraz (get the 3rd key fragment)
8. Old Hillsbrad Foothills (get attuned to Black Morass)
9. Black Morass (to combine the 3 key fragments into the Kara Key)

Here are the steps that you will need to take for these instances:

1. Fly to either Duskwood (Alliance) or the Swamp of Sorrows (Horde) and then ride to the Deadwind Pass. Here you will find Karazhan. (If you can't find it, Thottbot it for a map that will show you where to go.) Then, you go to the big building and talk to the NPC. He will give you 2 quests to do, accept both of them and go do them. After you get them done, he'll give you a third quest, which is the first part of the chain.
    
2. Fly to either Southshore (Alliance) or Tarren Mill (Horde) and ride over to where he sends you. (Once again, if you can't find it, Thottbott it.) This NPC will send you to Shatt'rath and the NPC there is in the middle of Shatt'rath. He will then give you the quest for the first key fragment, which is in the Shadow Labryinth.
    
3. **NOTE: this step can actually be skipped if you know someone with a Shadow Labyrinth key. If not, to get a Shadow Labyrinth key, you have to loot the key from the last boss in Sethekk Halls, but to be able to loot it, you need to have done the Mana Tombs. So, start with the Mana Tombs and then do Sethekk Halls which will get you attunded to Shadow Labyrinth.
    
4. Go do this instance and get the first key fragment then turn in the quest. Accept the next quest for the second and third part of the key fragments.
    
5. Go do the Steamvault and get the second key fragment.
    
6. **NOTE: this step can be skipped if you have a friend with the Arcatraz key, or a rogue with high lockpicking skills. If not, fly to Area 52 in Netherstorm and accept the quest there.
    
7. Go do that quest for the third key fragment.
    
8. Run through Old Hillsbrad Foothills in order to get attuned for Black Morass.
    
9. Combine the 3 key fragments into the Karazhan key after beating the Black Morass event.

You are now Karazhan keyed!