The trash before Peroth'arn isn't too bad. That first huge Legion Demon will charge at other party members, so as a tank, be prepared to wrangle him back. Once you're facing Peroth'arn, the two biggest things to look for are his Fel Flames and his eyes. Fel Flames will cause a small circle of flame to appear under you, becoming a patch of high-damage fire. As a DPSer, get out of that. As a tank, get out of it, and also pull Peroth'arn away from it so your melee DPS can keep DPSing without being blocked by the flames.
At about 70%, he'll use Drain Essence, then disappear and send out eyes after the party. Keep away from the eyes, because if one hits you, he will reappear and immediately attack you. That means run out to the perimeter of the ring if you need to. I have had some groups that have insisted tanks should let themselves be caught so as to take the brunt of the attack, but it should be possible to completely avoid the eyes.
His drops include the Girdle of the Queen's Champion, a pretty solid DPS belt with hit and critical strike rating. Convert whichever stat you need less to haste, and you're golden.
Queen Azshara's trash is made up primarily of spellcasters. While interruptions aren't mandatory, there's no harm in doing so. At the least, you'll want to interrupt the Royal Handmaidens' Sweet Lullaby.
Queen Azshara herself won't actually be targetable. Instead, you'll find her Enchanted Magi, which will come in groups of two. They're all casters, so they'll be somewhat hard to wrangle, but you can interrupt their basic casts.
As a DPSer, though, focusing on her adds is actually your tertiary goal. Your first goal is to interrupt Azshara's Total Obedience spell. It takes 8 seconds to cast, but if it casts, everyone's mind-controlled and Azshara will kill you all. She'll shout before she uses it, so when she does, run over and interrupt it at all costs. Of course, if you're lucky, you'll have a mage with Counterspell, but if you're unlucky, that mage will completely forget to cast it.
Your secondary goal as a DPSer on this fight is to destroy Hand of the Queen. Azshara will sometimes mind-control another member of the group with an ability that will put a giant hand on their character. When this happens, she will shout -- so again, learn the shout, respond, and kill that hand. Only after you do both these things should you go back to killing adds.
Azshara will drop the Cloak of the Royal Protector, which contains a nice chunk of haste, a great secondary stat for death knight DPS of all stripes. For tanks, you can get the Queen's Boon, a dodge and parry tank ring.
Mannoroth and Varo'then
This fight can get a little complicated, but you will have both Illidan and Tyrande helping you out. Malfurion's busy being a jerk, so don't count on him. The trash is pretty simple. You'll want to keep near Tyrande for the first pack and Illidan for the big abyssal. For the Shadowbats, keep your Death and Decay handy, as the Shadowbats will go immune to non-AoE damage at certain intervals.
For the fight itself, focus on Varo'then first, then Mannoroth. Tanks are also going to want to watch the adds. While Tyrande can take them out for the most part, they can get free and start harassing your healer, so if you can, try to get them wrangled in. In addition, Tyrande does have her moonlight up for certain phases of this fight, so you can stand in it (or have your ranged DPSers stand in it) to get extra damage against the demon adds.
In the Mannoroth phase, let Illidan tank him and focus exclusively on the adds. Also, once Varo'then goes down, be absolutely sure to grab his sword and hurl it at Mannoroth. I've seen people forget before, and since you'll be in melee range, you'll be right there to do it.
For both tanks and DPS, the big issue here is going to be dodging Fel Firestorm. If you've fought earlier Pit Lord bosses, you know the drill. Just keep moving so the fire doesn't hit you. Eventually, Illidan will give you his demon form buff. This should be the perfect opportunity to finish off adds, so don't be afraid to run around and immolate them down to a more manageable level before you face down Mannoroth himself. Still, with that health regen, you probably won't die anyway.
Mannoroth offers some pretty awesome rewards for the death knight. Pit Lord's Destroyer is a nice-looking two-handed mace with expertise and mastery, making it great for tanks especially and passable for DPSers. Varo'then's Brooch is a nice, solid strength trinket with a mastery proc. Finally, Annihilan Helm offers you tanking death knights relief if your tier helm token refused to drop in Firelands.
If you grabbed the quests from Alurmi and Nozdormu at the start of the instance, you'll also have some quest rewards to choose from. Specifically, you may want to pick up Treads of the Past, a set of DPS boots with somewhat middling secondary stats.
Learn the ropes of endgame play with WoW Insider's DK 101 guide. Make yourself invaluable to your raid group with Mind Freeze and other interrupts, gear up with pre-heroic DPS gear or pre-heroic tank gear, and plot your path to tier 11/valor point DPS gear.