Since we have a lot of new players in our guild I decided I would attempt to write a walk-through for the Karazhan key attunement quest chain. Not that the one on WoW Wiki isn't perfectly fine, of course.
If you want to follow mine, or any other Karazhan attunement guide, I would advise reading through the whole thing before you set off so that you are familiar with the steps you need to take to get the key. For example, if you read the whole thing you'll know you need to run the Caverns of Time: Durnholde (Old Hillsbrad Foothills) instance, but that it doesn't matter at which point you do. This information might mean you can easily slip into a group which is looking for one more, rather than getting to that stage and needing to make your own group from scratch.
The quest chain starts in Deadwind Pass (between Duskwood and Swamp of Sorrows), outside the Karazhan instance. An NPC there, called Archmage Alturus, will give you the quests Arcane Disturbances and Restless Activity. They both involve the same area, just under Karazhan. One requires you to get ten Ghostly Essence, the other requires you to take some readings of a well and a pond. To get the readings you need to go down two hatches near Archmage Alturus, one to the north of the other. You fight your way down the trapdoors and at the end of one underground area there is a well and at the end of the other is a lake. Once you've got your essences and readings you can return to Alturus.
The next step is to take Alturus' report to Archmage Cedric on the shore to the north of Dalaran (the huge candyfloss bubble to the south of Undercity and west of Tarren Mill in Alterac Mountains). This step is easy enough, just a little time consuming as you have to travel from Deadwind Pass to Alterac. These things are sent to try us and we must be equal to the challenge!
Once Cedric has played his minor part you are shipped off to Shattrath to speak with Khadgar (he is located in the middle of the Terrace of Light with A'dal and is the guy who made you follow his stupid elemental guide before choosing which faction in Shat you wanted to work with).
Khadgar will give you the quest Entry Into Karazhan, which requires you to go into the Auchindoun instance, Shadow Labyrinth. If you get a group to this instance and you fail to down Murmur, you can still get the key fragment as you wont agro Murmur if you just run around him to the left, get up on the ledge and open the container. Almost every single group I've ever been in has been more than happy to take out the arcane guardian so don't worry about not getting it done. Even if most of the group doesn't want to help you it's a really weak elite which can be soloed by the right classes. To get into Shadow Labyrinth at least one member of your group must have the Shadow Labyrinth Key or a rogue with a 350 lockpicking level. You can easily get the key from Talon King Ikiss in Sethekk Halls.
Khadgar will then send you off to find The Second and Third Fragments which are in the Steamvaults (Coilfang Reservoir) and the Arcatraz (Tempest Keep) respectively. The Steamvaults doesn't require anything special to get into. The second container is in at the bottom of the lake next to the room in which you fight the first boss. The Arcatraz is a little more tricky. You need at least one person in the group with the Key to the Arcatraz (or a rogue, again with 350 lockpicking), which you get by doing the quest chain starting in Area 52 with Consortium Crystal Collection and ending with How to Break Into the Arcatraz given by A'dal, which requires you to run the Botanica and the Mechanar. You will also need a flying mount to get to the Arcatraz, as you can't be summoned there. The container for the key fragment in the Arcatraz is in a room full of large void elementals, just around the corner from where you enter the room. Both the second and third guardians are easily soloable if your group clears the instance and doesn't want to help you get the fragments, not that that has actually ever happened to me.
Once you have completed that stage, Khadgar will send you off to the Black Morass in the Caverns of Time in Tanaris to get The Master's Touch. To gain access to the Black Morass you'll need to be attuned, which requires you to have completed the quest to free Thrall in the Old Hillsbrad Foothills instance in the Caverns of Time. Entrance into which requires you to have followed an NPC around the Caverns of Time (which you might want to do before you start looking for a group so you don't hold things up).
Once the master has touched your key (or rather, given you his own key) then you just need to return to Khadgar, slap him* for the hoops he made you jump through and complete the quest: Return to Khadgar. Congratulations, you now have your very own, shiny Master's Key and are totally attuned for Karazhan.
The Kara key attunement process can be a total pain. If you want to be self sufficient (as I did the first time I got attuned) then you must run eight instances and do a very long side quest chain for the Arcatraz key. The minimum number of five man instances you have to run is five (you don't have to get the Shadow Labs or Arcatraz keys, someone else will probably have them, not guaranteed, but likely) which is still a large slice of time.
* Surgeon General's health warning, slapping Khadgar may be a life limiting move and may lead to, amongst other things: lung cancer, heart disease, brain tumours, obesity, cirrhosis of the liver, skin cancer, kidney failure, kidney stones and fireballs to the face.