When you died your soul was grabbed by the Lich-King, Arthas, and you were made into an elite unholy warrior - a Death Knight. Eventually you and your associates broke away from his control and now you're on your own, to pursue your own goals.
As a Death Knight you are well equipped, literally, when you emerge into your new life. Your gear is better than most and your talents and abilities will allow you to crush the opposition and level quickly. When you hit the "end-game" (level 85) you will be able to do well in PvP as well as raids as either Tank or Damage. All in all, not a bad place to be.
Leveling your Death Knight is easy. As you go through the starting area you will gain the gear and your talents as you complete quests. Pay attention to the quest info and you should have no problem completing everything. Be aware that some quests may take awhile to complete, if you're unlucky, so just be patient.
You'll be level 57 when you're all done. If you participate in a little more destruction while leveling you'll be 58. People using Blizzard's Recruit a Friend system (3x Experience awarded) will be about level 62. When it's done your next trip will be to the capital city of your faction (Orgrimmar or Stormwind,) where you will brave the insults, rotten fruit, and death threats as you travel to your leader and present yourself. Next you either check your leveling guide or head to the Command Board near the entrance to the city. It will tell you where to go for your next task.
Which race should you be?
Any race can be a DK, and they all can be effective and all have some racial ability that will help in some area, whether it's PvE (Player Vs Environment) or PvP (Player Vs Player.) Here are some suggestions if you want maximum efficiency:
If you'd rather play a Blood Elf or a Gnome then go for it.
Which Specialization (Spec?)
Like every other class in the World of Warcraft, Death knights have three specs, in this case those are: Blood, Unholy, and Frost
Then you also have three "stances," much like the Warrior "stances," that modify what you do. Interestingly, for patch 4.3, the Frost spec does the most damage while in Unholy stance (it's not a big difference, though.)
Stats and Gear
For all of your leveling, PvP, and end-game experience Strength is your #1 stat. If you come across any gear that has Int, Spirit, or Agility then sell it or give it to the other guy, it's not for you. Any gems or enchants that have Strength are much more valuable, point for point, than those that don't.
For leveling and dungeons/raids you will want some Stamina and for PvP or Tanking you will want a lot.
Resilience reduces damage from other players and is your second most important stat in PvP. Resilience has zero value outside of PvP.
Basically you're going to share the same gear as Warriors and Paladins (not the healer Pallies, though.) Plate armor with Strength and Stamina, and Resilience if you're in PvP. You can use one or two-handed weapons and you can dual wield the one-handers. You cannot use a shield or ranged weapons, but your abilities will make up for that loss.
Which secondary stats do you need?
For any situation that does not involves Tanking you will want Hit Rating (up to 8%) to make sure you hit with your main strikes. Haste is a generally good stat for Death Knights. Frost makes good use of the Expertise stat, but for Unholy Expertise is the weakest stat of all.
Tanking is a bit different. With the changes in patch 4.3 threat shouldn't be an issue, so damage mitigation becomes the top priority. You'll want to stack Stamina and then Mastery and Dodge as your secondary stats. Gearing for damage mitigation willmake things a bit easier on your healers and help get through massive damage spikes.
Runes
Death Knights use both Runes and Runic power to drive their abilities, in much the same way that Paladins have both Mana and Holy Power. Each of your abilities will use one or the other. The abilities that use runes will generate some amount of Runic Power. Death Strike, for example, costs one Front and one Unholy rune to use and generates 15 Runic power when used. Various talents will modify Runic power generation, rune usage, and so on.
In addition to all of that you get a ghoul for a pet. This "cute" little guy only hangs around for a couple of minutes unless you're in Unholy spec, then he's like any other combat pet and just as useful.
All in all the Death Knight is a powerful, interesting class with a lot of capabilities. You do have to reach level 55 with some other character before you can create your first Death Knight, but that's not particularly hard. Give it a try and crush your opposition.