I had to explain this to a fellow guildee last night, so I though it would make a good post and great information to any tank coming up on the LK fight.
First, you have to decide what role you're going to fill; Main tank or off tank. And this decision actually has a lot to do with what class you are, and how good of a player you are.
You want situational awareness. There a LOT going on during this fight, and you've got to keep track of a lot as one of the tanks.
PHASE 1 (LK 100% HP)
Main tank engages LK and positions him. LK will spawn wave after wave of ghouls. It's your job to pick up and hold aggro on these as well as the LK.
Offtank picks up the Shambling Horrors as well as one or two of the Ghouls.
Main mechanic - Necrotic Plague - stacking dot that thins out the ghouls and horrors. The plague jumps targets, and as it does it gains a stack becoming more powerful. It's up to the raid to manage the stacks and make sure it does not fall off. It's also the job of the healers to make sure this DOT does not tick on a player as it will kill them instantly (you have only a few seconds to get into the proper position for it to be cleansed off you and it jumps to a new target).
TRANSITION PHASE 1 (LK 70% hp)
Everyone moves to outer ledge, finish off horrors and ghouls if you have any, and then lose the disease by having a player get out of range of everybody else and cleanse it off. Here's where it gets tricky for the tanks. The spirits that start to spawn do a conal silence effect. This means your tanks are going to be silenced - this also means they cannot taunt. We chose to bypass this by having the main tank stand off to the side out of the cone and taunt mobs to the off tank. This allows the main tank to freely engage the LK at the start of phase 2 with no spirits also beating on him.
PHASE 2 (LK 70ish %)
This is the hardes phase, due to the RNG factor of Defile placements and well as valkyr paths.
Main tank engages LK and the entire group moves to the middle. Off tank must be careful with positioning as you will likely have spirits up still. Make sure you don't silence your raid. A Valkyr will spawn and select a target - all dps must move to kill the Valkyr before she drops a fellow raid member off the platform. Stuns and snares DO work, make sure you apply them.
During the Valkyr, one raid member will be targeted for Defile, which puts an aoe where ever the player happens to be. As the main tank has the LK targeted, it's help full to call out who's going to get hit, and hopefully people can move away from them ASAP.
As a tank, the main thing you need to be concerned with during this phase is placement of the LK as close to the center as possible with out getting in defiles, as well as dealing with Soul Reaper. This deals 50.000 shadow damage to the tank and increases the LK's haste by 100% for five seconds. If you're not prepared for it, you will more than likely get instagibbed. Blow what ever cooldown you have to live through this.
I find as a warrior tank that you can take 3 Soul Reapers before you need to worry about having your off tank pickup the LK. On the first one, blow Shield wall, second pop Last Stand, and on the third pop your corroded skeleton key (4 piece tier 10 bonus works too) and enraged regen. After which you will be out of cooldowns and should call for the off tank to pick up the LK right after you get the fourth Soul Reaper (you'll take the damage of the debuff, but will not have to worry about surviving the haste buffed LK).
You are a one man army. This is all you! DKs are perhaps the best suited tank to deal with Soul Reapers. If you're a good little DK and are specc'd into the blood tree, you're golden on this fight. Dealing with Soul Reapers is cake, because you always have a cool down up and don't have to worry about another tank taunting. You can chain your Vampiric Blood, Anti Magic Shell, and Icebound Fortitude and have them come off of cooldown in time to continue popping them. (It's rather OP that the Anti Magic Shell makes you immune to the shadow damage suffered by Soul Reaper)
The off tank in this phase should help kill the Valkyrs after his spirits are dead.
TRANSITION PHASE 2 (LK 40%)
Everybody off the platform again and get into the same setup as on the first transition phase, where one of the tanks taunts spirits to the other one.
PHASE 3 (LK 40 ish %)
Main tank drags LK to the complete edge of one side of the map, and the off tank drags any remaining spirits on the platform and finishes them off with the DPS.
Main tank waits for vile spirits (Can't miss it, LK stops and raises Frostmourne and spirits come out of the sword.) and then hauls ass to the complete opposite side of the map. Off tank should run around and try to soak up any of the spirits heading for raid members. Rinse and repeat until LK is down to 10% at which point he kills all of you, and then you get your free kill after being resurrected by the NPC.