The timed run is probably old news for some players, but surprisingly there are still quite a few who do not have it yet. I am going to write a quick guide on how to accomplish this. It will be from a Warrior tank's perspective but should help out classes of all types.
This instance can be a bit stressful if either the tank or the healer is undergeared. The following is a rating of importance, on a scale of 1 to 10, for each type of class (and the corresponding difficulty ratings).
Tanking: 9 out of 10.
Bad tanking is one of the things that can make the timed run go down hill in a hurry. It takes skillful chain pulling while keeping a close eye on your healer's mana bar (as any good tank should be doing anyways). You also need to make sure that all elite mobs are beating on you. The little mobs respawn very quickly and will keep putting you in combat, making ressurecting difficult at times. Keep the mobs off of your healer and dps. Time rezzing is time better spent killing.
Healing: 7 to 9 out of 10.
Regardless of the tank, this instance is not usually a cakewalk. With a well geared tank who knows their job, you will have to be on your toes, but should not be going out of mana. An undergeared tank plus overgeared DPS can make you job much harder as they will probably be yanking aggro.
DPS: 5 to 6 out of 10.
Despite needing at least decent DPS, the timed run can be completed with fairly mediocre dps. 1500 is probably the minimum for each DPS, which is pretty easily achieved. I ran one earlier in the week with all three DPS hovering around this number and finished with a minute to spare. I did it again with all three dps doing over 2k and finished with 6 minutes to spare. Dead DPS does no DPS. Staying alive is the most important thing.
The Instance: The instance starts with you revealing a bunch of plague crates and heading up the road where you see Jaina, Uther, and Arthas having an arguement. Arthas wants to destroy everyone, Jaina and Uther QQ and leave. You talk to Arthas to start the instance. Once you talk to him, run all the way to the T - intersection straight ahead of you. Eventually, the first elite mob will pop up to the east of you (or right if you are facing the the end of the T - intersection).
You will be AOE tanking all of these mobs. Choose the most dangerous mob and make sure to focus on them. I mark my first target with a skull when I charge in. After the first elite mobs, they spawn at random spots along the streets in Stratholme. There will be an arrow on your mini map pointing you where to go. Each time you down a group of elites it will pop up to point the way. Be quick about it. As soon as a group is done you should be on your way to the next.
As I get to each mob, I will charge in and thunderclap to get initial aggro. I will then take a step back and shockwave the mobs. All of the elite non-boss mobs are stunnable. This stun should allow you healer time to get into position. You should thunderclap and cleave constantly along with your normal rotation. Throw in a demo-shout as well to make healing a bit easier. After 5 waves, you will be greeted by the first boss of the instance, the abomination Meathook.
Now is a good time to let your healer drink briefly if they need to. To maximize your time, you should probably be needing or greeding everything which means not passing. Meathook is basically a tank and spank fight. I believe he puts a dot or something on a party member which incapacitates them and does significant damage. It is not a terribly difficult fight. From a tank perspective it is tank and spank. Once he is down you will have a moment to regroup, drink, sort loot if needed before the next pulls.
You will do several more gauntlet type pulls until the next boss. On the trash mobs, I prefer to kill in order, Abominations, Crypt Lords, Caster mobs, then the Ghouls. The next boss will be Salramm the Fleshcrafter who looks pretty much like Heigan the unclean. From a tank perspective, it is basically tank and spank. He will be tossing shadowbolts that can be reflected back at him. He will also remove flesh or something like that which I am not completely sure about. Look it up.
Once he is down, Arthas should run to the Inn where you will have to talk to him again. Run ahead of him and mana up before he gets there. When he arrived, talk to him again to start the next gauntlet. It will start with a group of 3 Inifinite Dragonflight. In pretty much every pack of infinite dragonflight there will be caster. Heroic Throw at them to get them in range. After the first pack, you will run up stairs and be met by three more backs of about 4 infinite dragonflight before the next boss.
Chrono-Lord Epoch from a tanking point of view is pretty much tank and spank. He will stun you from time to time and will do a "time warp" where he will charge random party members. Burn him down and Arthas will reveal a hidden doorway that will take you down to the street.
Once you get to the street, you will need to talk to Arthas to start the next part of the gauntlet. You want to pull quickly, but try not to overwhelm you healer. Keep a look out for patrols wandering closer to your location. If they do, be quick to taunt and start a rotation on them to keep them off of your healer and dps.
Once you get to the end of the gauntlet, you will have to race to the timed boss, who I think is called the Infinite Corrupter or something like that. If you have a lot of time left feel free to tank him where he stands. If you are coming down to the wire, pull him with Heroic throw and tank him backing up. You will want to kite him down the way you just came. This will give the DPS that much extra time to burn him down if the timer runs out. He will have to run to a portal where you originally inititated combat with him and can be killed during that time.
Once he is dead, loot him and get your shiny new Bronze Drake!
Then head over to Malganis and give him a nice thrashing and hopefully get your shield (for looks if for nothing else, the thing looks damn cool).
A side note. Make sure you are using your defensive cooldowns. Shield wall has saved my ass on these runs a few times. Dont find yourself dead with all of your coolddowns still up. Be a tank stud, not a tank dud :)