At 68 head to Deadwind Pass outside of Duskwood and follow the path past the ogres to Karazhan. There’s a guy with an exclamation point - he's got two quests, Arcane Disturbances and Restless Activity. Both are a little ways away in the caverns under Karazhan. Just look for steps leading down to the basements. Once completed, he sends you to Dalaran. The guy in Dalaran sends you to Khadgar in Shattrath.
Khadgar is the guy hanging out in the middle of Shattrath - the one who gave you the Shattrath tour with the pink voidwalker. He gives you The First Key Fragment, and finally, you have the actual Karazhan attunement chain.
Okay. It sounds easy. Go into Shadow Labyrinth, get the first key fragment, turn it in… done. But it’s not that simple.
While you are finding a group for Shadow Labs, head out to Area 52 in Netherstorm and get the Assisting the Consortium quest. You need to finish the entire chain in order to get the key which will enable you to enter the Arcatraz, which you will need to finish the Karazhan key fragment key chain.
If you get tired of Netherstorm (and if you're still 68, dying in Netherstorm) head out to Honor Hold or Thrallmar and get Key to the Citadel quest line. The chain will take you to Shadowmoon Valley and eventually give you the key to Shattered Halls. Shattered Halls is pretty much the fastest way to get Honor Hold / Thrallmar to revered rep, which you will need to get the Heroic key. Heroics are needed for later attunements, so it's best to start early.
Once you complete Shadow Labs, turn The First Key fragment into Khadgar and get the second and third fragment. The second fragment is in Steamvaults in Coilfang Reservoir. This is a 70 instance but the fragment is not hard to get. It's before any of the bosses so if you can find a group willing to clear to the fragment for you before you hit 70, go for it. If you are 70 and are a class that can stealth, you can stealth to it (you'll need at least two though). If you can't stealth, an invisibility pot and a soulstone and you can run to a safe spot, die from the aggro, rez, and get the fragment (again, though, you need two people to fight the spawn). Where's the safe spot? Well next to the tunnel leading to the second and third boss there's a steep hill. Up at the top of this hill is safe. The fragment is in the pool below.