Kael and his soldiers were sent to Silverpine Forest, where they began to notice disturbances around the city of Dalaran. They packed up and prepared to back into safer territory when Kael ran into Maiev Shadowsong and Tyrande Whisperwind. Kael told the two night elves that he thought Illidan might be behind the problems at Dalaran, and both offered him their assistance. On the way to Dalaran, the night elf/blood elf caravan was ambushed by undead. Maiev and Kael led their troops across a bridge to safety, while Tyrande stayed to fight off the undead and was swept away by the river. Kael started to go after her, but the big jerk Maiev told him that he needed to stay and help her catch Illidan.
The duo continued to Dalaran and met up with Malfurion, who was understandably upset that Tyrande had been swept into a river and presumably killed. They interrupted Illidan's big magic ritual, which would actually have killed the Lich King and probably saved Azeroth a lot of trouble in the end. Malfurion yelled at Illidan for his part in Tyrande's death, and Kael noted that they didn't actually see her die. Malfurion and Illidan took off to save the girl they both loved, closely followed by Maiev, while Kael'Thas stood in the ruins of Dalaran (probably vowing to swear off dating for life.)
Kael'Thas met up with Grand Marshal Garithos, the commander of the Alliance at Dalaran and vehement racist that was one of the most singularly unpleasant people in the Warcraft universe. Kael wanted to help Garithos fight off the undead, but instead, the marshal sent them to repair some arcane observatories across Lordamere Lake. Unfortunately, Kael's shipyard was burned. Lady Vashj and her naga offered him some boats free of charge, and he took the offer and went to repair the observatories.
Garithos totally freaked out when he heard that Kael had accepted help from the nagas. Garithos claimed they were a threat to the Alliance and a bad race of people and ... actually had a point there, all things considered. As some form of revenge, Garithos sent Kael to fight an undead force in the west with nearly no troops. Vashj and her naga saved the day once again, as a combined elf/naga force crushed the undead. Afterwards, Vashj and Kael discussed their peoples, and Vashj revealed that the blood elves would suffer a fatal magic withdrawal without the Sunwell. This private conversation was interrupted by the arrival of Garithos, who arrested Kael and his elves for treason for consorting with the naga.
Kael and his troops were imprisoned in the dungeons of Dalaran, but were soon freed by Lady Vashj. She promised Kael and the elves a new life in Outland if he'd only follow Illidan through a dimensional portal. I can imagine his thought process at this point. "Well, on one hand, Illidan is pretty evil and I would be abandoning everything I ever cared about to go to a hellish wasteland. But on the other hand, human guards are coming at me with long pointy things. Portal time!"
Once in Outland, Kael and Vashj helped free Illidan from Maiev. Illidan thanked Kael and told him that he would help the blood elves with their magic addiction in Outland. Kael sent one blood elf, Rommath, back to Silvermoon City with the good news, and then aided Illidan in taking Outland for his own. When Illidan was called back to Azeroth by Kil'Jaeden to fight the Lich King, Kael gladly went along, hoping for some revenge on the Scourge. Kael and Arthas faced off several times, but neither managed to kill the other. In the end, Arthas smacked Illidan down and became the new Lich King, while Kael and Vashj picked up their master and dragged him back to Outland to recover.
Where he is now, part I: Unlike most lore figures, Kael'Thas has been very active during the time of World of Warcraft. He set himself up as "King Kael'Thas" in Outland, and moved into a beautiful magic citadel called Tempest Keep. A renegade group of Draenei stole part of the Citadel and crashed it into Azeroth, where it now serves as the Exodar. Kael'Thas was not pleased by this turn of events, and has detailed plans to send blood elves through a gate from Tempest Keep to take the Exodar back. Kael also led an assault on the Naaru stronghold of Shattrath City, but failed when a group of blood elves deserted him (later becoming the Scryers.)
NOTE: SPOILERS FOLLOW, IF YOU'RE AN ALDOR.
Why we have to kill him:
Illidan has gone a little wacko in Outland, and while Lady Vashj is sticking by him, Kael'thas seems to have weighed his loyalty to Illidan against his loyalty to himself and gone for the second option. His Sunfury legions of elves still serve Illidan in Shadowmoon Valley (while Illidan pretends to serve the Legion but secretly trains for an anti-demon war), but Kael has set up his own little operation in Netherstorm. He's built a ton of mana forges that harvest energy from the Twisting Nether, but tear apart the land itself. He's also apparently taken leave of his senses and allied with the Burning Legion for some nefarious purpose. Some suggest that in the end, Kael will also turn on the Burning Legion, and drain their magic energy to help the elves.
There's a more pressing problem right now, though. Kael'thas has discovered the Cipher of Damnation, which was used by Gul'Dan to separate the Orcs from the spirits of Draenor and may have summoned Ragnaros to Azeroth. Therefore, we have to go to Tempest Keep and keep him from ever using this spell!
Where he is now, part II:
There's also been some really interesting speculation about what exactly Kael'thas is planning. Why did he ally with the Burning Legion? What are the manaforges being used for? Why doesn't he seem to care that his actions are destroying the Netherstorm? Master lorekeeper Ghostwalka has some clues (read down to post 6.) Basically, Ghostwalka suggests that Kael is using the manaforges to gather magic energy for Tempest Keep while bringing more new blood elves to Outland. When he has enough magical energy, he will betray both Illidan and the Burning Legion and use this energy to shift Tempest Keep across dimensions to Silvermoon City -- destroying the remainder of Outland in the process. Thus, King Kael would come back as a hero to his people, and he will be finally able to sate their magical addiction by planting a new Well of Eternity with the vial that Illidan gave him. Oh, and hey, remember the Scourge that killed Quel'Thalas and the humans that imprisoned the blood elves? Revenge time.
This is purely speculation on Ghostwalka's part, but one can note that his original post about Kael'Thas had his article, and all quotes of it, deleted by Blizzard. Hmm ... speculation, or inadvertent spoilers?
Identifying characteristics: Blond hair, green eyes, red dress - I mean robe. Deep hatred for the Scourge and the Alliance. He relies mostly on his magic for fighting, but also wields the Flamestrike sword. It's known that he summons pet adds, and may have an AOE fire attack.
For more information: Warcraft Encyclopedia on Kael'thas, WoWWiki on Kael'thas, Ghostwalka's "The Fall Of Prince Kael" in his Studies on the Races, the original response to Fall of Prince Kael.