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.
What he's up to now: While players haven't encountered Deathwing for some time, he's been quite busy. According to Night of the Dragon, Deathwing has been hidden away from the prying eyes of the other Aspects and attempting to create the ultimate Dragonflight with which to conquer (or destroy) Azeroth. His son, Nefarian, was unsuccessful in creating the Chromatic Dragonflight army, thanks to player intervention, but Deathwing has learned from his son's failure. The Netherwing flight of Outland is physiologically different from other drakes, the result of Black Dragonflight eggs being exposed to the twisted energy present on the chaotic planet, and Deathwing has discovered their unique powers. Night of the Dragon sees Deathwing successful in his attempt to fuse Black Dragons with Nether Dragon essences, resulting in the dangerous Twilight Drakes seen in Wyrmrest Temple's Obsidian Sanctum. While players are successful in destroying the clutches of eggs there, as well as their keepers, Sartharion and his drakes, Deathwing has access to other eggs of all of the various flights in Grim Batol. His artificially-created leviathan of a son, Dargonax, was also destroyed, along with his prime consort Sinestra. But Deathwing was never truly defeated, and nor did he ever rest.
Having retreated to the elemental plane of earth, Deepholm, Deathwing planned his full and devastating revenge against not just his fellow Aspects, but the entire planet of Azeroth. Erupting through the ceiling of Deepholm, Deathwing brought about the Great Cataclysm -- forever changing the face of Azeroth and its people. Bringing untold destruction in his wake, he took flight to the Red Dragonflight's former prison, Grim Batol, and began to calculate the true end of all living things on Azeroth -- possibly sparing the Twilight's Hammer Clan, who now serve only him.
Identifying characteristics: Neltharion/Deathwing is a gargantuan black dragon with a molten body, barely held together by the adamantium armor that must constantly be refitted to keep him from melting.
Why you've spent way too many hours fighting him already: Onyxia and Nefarian are the daughter and son, respectively, of Deathwing. Onyxia is posing as a human counselor to the king of Stormwind in hopes of destroying the Alliance. King Varian Wrynn uncovered her plot and slew her. Nefarian was able to enslave the orcs of the Blackrock Clan and set up a stronghold in upper Blackrock Mountain. From there, he tried to kill Ragnaros and then take over the world. The Netherwing Flight in Outlands are also children of Deathwing, being black dragonflight eggs affected by Outland's chaotic energies. And while it may not be obvious due to lack of in-game reference, the Twilight Flight in Northrend's Obsidian Sanctum is in service of Deathwing, being the result of his mixture of Nether Dragon essence with Black Drakes.
For more information: WoWWiki on Deathwing/Neltharion, Neltharion on Blizzplanet, or read Day of the Dragon and Night of the Dragon, both focusing on Deathwing.