Gul'dan summoned his loyal necrolytes to a meeting, and then proceeded to sacrifice them all. This is why you don't make deals with demons -- the retirement plan sucks. Gul'dan used the sacrifice to power an enormous bit of magic -- he transplanted the souls of the Shadow Council members Doomhammer had killed into the bodies of dead Knights of Stormwind. He gave his new Death Knights jeweled truncheons filled with the powers of the dead necrolytes. Doomhammer liked the new Death Knights and ordered them to go to different orc clans -- however, they were still loyal to Gul'dan at heart. Doomhammer did not think this one through. How are you going to trust beings that are a) the spirits of people you killed b) in the body of your sworn enemies c) given the powers of people who plotted against you d) by someone who hates you?
While most of the Death Knights were killed when the Alliance won the Second War, a few snuck back across the Dark Portal with their leader, Teron Gorefiend. Gorefiend and the Death Knights pledged their loyalty to the elder shaman Ner'zhul in return for a world of their own. They led attacks on Azeroth to pillage valuable magical artifacts, since Ner'zhul wanted to open portals to other worlds as a way for himself to escape the wrath of Kil'jaeden.
However, the spell got messed up and Draenor went kablooie. Most of the Death Knights died, although some of their spirits still haunt Shadowmoon Valley (and possibly Deadwind Pass.) Teron Gorefiend was also killed, although if you've done a certain questline in Shadowmoon Valley, you helped him come back from the dead a second time and join Illidan in the Black Temple, where he's creating a new order of Death Knights from fel orcs. Shame on you! Now go to the Black Temple and kill him. Make sure to bring 24 of your friends and a full night's worth of buffs. And sandwiches. Illidan loves sandwiches.
So let's fast-forward twentyish years.The original Death Knights are dead, but the Burning Legion is threatening the world again through the transformed spirit of Ner'zhul, known as the Lich King. Prince Arthas went to find the mystical runeblade Frostmourne so he could rid the world of the undead once and for all. Unfortunately for him, the runeblade stole his soul and made him loyal to the Lich King. Upon swearing his loyalty, the former paladin lost the blessing of the Light and gained necromantic powers.Arthas became the first Death Knight in years, and the first ever to become a Death Knight without ... uh ... dying.
Like many converts to a new religion, Arthas thought that being a Death Knight was so cool, he had to share it with his friends. Other paladins went to Northrend to find the source of the plague of undeath, and along the way, they also became embittered and corrupted. It was easy for Arthas and Ner'zhul to convince them to come over to the dark side, and grant them a rune blade that gave them necromantic powers. The new Death Knights led the forces of the Scourge, and even today live on as trash mobs in Naxxramas.
So how do these guys get from serving the Lich King to working with the Alliance and Horde? Good question. A lot depends on the quest chain you have to do to unlock the Death Knight class. The Wrath of the Lich King trailer shows a character dying to the undead and being resurrected as a Death Knight, but since the developers have said that becoming a Death Knight won't wipe out your old class, it can't work like that.
Presumably, your character will have to first become corrupted by darkness, letting you create the Death Knight. Then, either as your old class or as a new Death Knight, you'll have to find a way to turn from the Lich King and help your faction -- and convince your faction your intentions are real (no srsly guyz!) Given that both the Old Horde Death Knights and the Scourge Death Knights are detested by both factions, this will not be easy.
Distinguishing characteristics: Blizzard has said that the Death Knight class will use a lot of the abilities from Warcraft III. Death Knights will be a tank/DPS hybrid, able to tank with two-handed weapons or dual-wielding, but only using bladed weaponry. Instead of mana, rage or energy, they will use runes of three different types (blood, frost and unholy) etched into their blades. The runes will be used up by abilities and will regenerate after a certain cooldown. The talent trees will also be Blood, Frost and Unholy. They'll be able to summon undead minions and use debuff spells. Ah, but how do they look? Hard to tell, but I like to think they'll look like the gothy students in Scholomance.
For more information: WoWWiki on Death Knight Lore, DeathKnight.info, Unholy Aura, WoWWiki on the Death Knight hero class.