History: Neltharion was the dragon aspect of the earth and the deep places of Azeroth. Along with his fellow aspects, Alexstrasza, Ysera, Nozdormu, and Malygos, he was created and given power by the Titans so he could watch over Azeroth. At first, he did a good job. But then he discovered the prison of some of the Old Gods of Azeroth, and they began speaking to him. They convinced him that if he destroyed all the other races on Azeroth, he could rule the world with his dragonflight and keep Ysera and Alexstrasza as what would amount to draconian sex slaves.
The Old Gods convinced Neltharion to create an object called the Dragon Soul. With the help of goblins, he forged a great golden disc deep within the earth. Neltharion put part of his own blood and the essence of a demon of the Burning Legion in the disc, and then shielded it magically so the other dragons couldn't see what was inside. He then convinced the four other aspects to donate some of their powers to the Dragon Soul, telling them it would be a great weapon to defeat the Burning Legion.
All the dragonflights headed for the city of Zin-Azshari, where the night elves and the Burning Legion's demons were fighting it out. As promised, Neltharion channeled his power through the Dragon Soul to kill most of the demons. However, it also killed most of the night elves. Then Neltharion turned to the other dragons and essentially said, "By the way, I was the only one who didn't put any of my power in here, so I'm now stronger than you. And you know what would be awesome? Me ruling the world." The other dragons attacked him, but he used the Dragon Soul to kill most of the Blue Dragonflight and paralyze everyone else, except Alexstrasza, who Neltharion kind of had a thing for. Neltharion's appearance changed from a normal dragon into a big fiery demon dragon, and he was reborn as Deathwing the Destroyer.
After this battle, the four other aspects hid themselves, along with their dragonflights. The Dragon Soul (now known as the Demon Soul) was killing Deathwing, so he had the goblins forge him some adamantite armor to keep him from being torn apart. But before he got a chance to use the Demon Soul again, Malfurion Stormrage snuck into Deathwing's lair through the Emerald Dream and stole the Demon Soul. The Demon Soul went on to have many great adventures with Malfurion and Illidan that we're not going to get into here, but after the defeat of the Burning Legion the other four aspects sealed it up and Malfurion hid it so that Deathwing couldn't use it anymore.
Deathwing waited for ten thousand years, causing a lot of earth-related havoc in the meantime. Near the end of the second war (the one with the Horde attacking everything), Deathwing snuck across to Draenor and hid some eggs, which became the Netherwing Flight. After he got back, he discovered the location of the Demon Soul. He decided that if he couldn't use it to destroy the world, then someone ought to. He used a goblin servant to give the artifact to the orcs of Grim Batol, who used it to enslave Alexstrasza and the Red Dragonflight. He then tried to maneuver a human mage, Rhonin, and some of his companions in a complicated plot to get the orcs to move Alexstrasza and her eggs out of Grim Batol. There, he figured, he could capture or kill Alexstrasza and steal some of her eggs to raise as his own. He also posed as Lord Daval Prestor and became King of Alterac, in sort of a secondary plot to destroy the Alliance from within. But in the end, Rhonin destroyed the Demon Soul and Deathwing was sent running from the four other dragon aspects, now returned to their full power.
Where he is now: No one knows, not even his children. Since the first rule of Warcraft is that no one missing or dead ever actually stays missing or dead, we'll probably see him in the near future. He may have returned to his lair in the Outlands, but it's a pretty slim chance. He may also be in disguise as a human in Gilneas or in a new lair in Undermine or Mount Hyjal. The biggest probability is that Alexstrasza is keeping him captive in Grim Batol, which is why no one can get near there unless you're a Hordie trying to get your Onyxia key. Speaking of which ...
Identifying characteristics: Neltharion/Deathwing is a big black dragon with molten eyes. His heart is apparently fiery and visible through his skin. He's wearing a ton of adamantium armor, Deathwing's powers are over fire and earth, so he spits lava at his enemies and can cause volcanoes to erupt. No connection with Ragnaros, but they could probably throw some pretty excellent parties together.
Why you've spent way too many hours fighting him already: Onyxia and Nefarian are the daughter and son, respectively, of Deathwing. They inherited the family tendency to try to take over the world, although they went about it in different ways. Onyxia is posing as a human counselor to the king of Stormwind in hopes of destroying the Alliance, although every week she gets revealed and sent back to her lair as a dragon and then killed multiple times. She also manages to carry a backpack made of her own skin. Nefarian was able to enslave the orcs of the Blackrock Clan and set up a stronghold in upper Blackrock Mountain. From there, he tries to kill Ragnaros and then take over the world. Interestingly enough, Nefarian carries a tear of his father's that your mages have probably taken to wearing. Both Nefarian and Onyxia wonder why no one ever visits them anymore. Oh, and the Netherwing Flight in Outlands are also children of Deathwing, although significantly less evil.
For more information: WoWWiki on Deathwing/Neltharion, Neltharion on Blizzplanet, and a summary of the novel Day of the Dragon, which is all about Deathwing and should be available at your local bookstore.