The death knight treasure hunter's survival guide
If you have had a chance, try running the solo instance. The keys to get in can be gained out of the Arcane Troves you get at the end of your dailies, or if you're lucky, from the rare mobs scattered about the island. Once you talk to Taoshi and get inside, you'll notice that most of the traps and pitfalls involve slowing you down, stunning you, and various other ways to trip you up.
Luckily, as a death knight, you have a few special tricks to keep from getting tripped up in there. Anti-Magic Shell is your friend when you come up against those lightning pillars. Anti-Magic shell completely absorbs the lightning, thus also allowing you to avoid the slow effect that comes attached to it. Icebound Fortitude also comes into play here. Immunity to Stun effects can help if you can't get out of the paths of the giant mogu statues in time. Even summoning a ghoul can be useful in the first part, as it should be able to kill those annoying spiders in a few hits.
Self healing abilities are also going to see some play here. Unless you're an absolute ninja, you probably will eventually take damage from a trap or mob, so having some instant self-healing like the Lichborne trick, Death Pact, or Rune Tap will allow you to carry on and grab more chests without stopping and wasting valuable time on food or bandages.
On the talent side, you'll definitely want to keep Death's Advance handy. Not only will a burst of speed help for getting to the next set of chests in time, but the ability to keep snare effects from slowing you down will be helpful for obvious reasons. If you can get to the section of the palace with the patrolling mobs, Remorseless Winter can help you with some emergency crowd control. Freeze the mobs attacking you and loot that set of chests before your timer runs out. On the other hand, Gorefiend's Grasp can help you pull mobs out of the radius of those traps and into a place where can more easily knock them all down and get back to chest looting. Asphyxiate could also help with a quick stun and grab, but then you'd be giving up Death's Advance, so you'd want to weigh your options carefully.
Of course, you'll still probably want to round out your arsenal with some of the special items you can buy with Elder Charms of Good Fortune. Arcane Propellant and Sleep Dust or Frost Rune Traps for mobs are always top on my shopping list. But as a death knight, you definitely shouldn't forget the tools you already have handy. They can make you that much richer by the end of your run.
The pitfalls of daily questing
Running around the isle and getting your dailies done is pretty straightforward, but you'll want to keep your abilities in mind anyway. One ability that I've gotten a lot of use out of is Path of Frost. Now, admittedly, there's not much open water on the island itself, but I have sometimes been able to use a combination of Goblin Glider and path of frost to jump off a cliff and away from a bad pull. Where Path of Frost really shines though is when you use it with Glyph of Path of Frost. The Isle of Thunder really feels quite a bit more hilly than previous quest hubs, so there's a lot of places where dropping down from a cliff to grab that last quest mob will be a lot faster than going around. Luckily we have our budget safe fall move.
While most of the quest mobs are soloable by a raid finder geared death knight, be wary of a few exceptions. You can still solo them in theory, yes, but you will have to pull out all the stops. The Mighty Devilsaur so far is one of the biggest. His acid spit and venom breath will kill you pretty quickly if you don't get out of their way. Even Anti-Magic Shell won't be able to protect you from all of it, so this is another case where Death's Advance can let you get out of trouble and wait for another opening. You will also find your extra powers work well when you have to kill Nurkala, who comes with a rather large retinue. Interrupting the Prophet's heals and keeping out of the way of the Firespear and Frostspear's attacks (or healing up when you fail to dodge them) is much easier when you have the death knight's toolkit.
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.