While approaching the camp of the Free Folk in Skirling Pass, Jon Snow witnesses this giant building a tent from the bones and tusks of a mammoth. As Jon stares at the giant in disbelief, Ygritte warns him that giants are shy, but quick to anger and tells him how she once witnessed a giant smash a man into the ground like a hammer to a nail. This giant catches Jon gawking at him and grunts in annoyance. 
During the Battle of Castle Black Dongo emerges from the Haunted Forest holding a bow with the other members of Mance Rayder's army. He escorts Mag Mar Tun Doh Weg and his mammoth to the outer gate of the Wall. While a wildling bowman fails to reach the black brothers on top of the wall with his arrow, this giant crouches next to him, pulls a massive bow out and shoots a ballista bolt at the wall, breaking havoc on their defenses. His next arrow hits a brother with such force it causes him to fly off the other side of the wall and land in the middle of Castle Black.
At the gate, while Mag turns around the mammoth and several wildlings tie ropes to it, Dongo violently exposes the gate's supports and ties the other end of the ropes to them. While the mammoth tries to pull the gate open, men of the Night's Watch throw down flaming barrels of tar, which explode and drive the mammoth off. The giant chases after it, but this brings him within the firing arc of a large ballista mounted atop the Wall. The ballista bolt impales him from behind and he is killed, which enrages Mag and spurs him to lift the gate and confront the men within all by himself.
When Jon leaves Castle Black to kill Mance Rayder, Jon walks past Dongo's corpse, which has crows picking at its flesh.
|Season Three appearances|
|Valar Dohaeris||Dark Wings, Dark Words||Walk of Punishment||And Now His Watch Is Ended||Kissed by Fire|
|The Climb||The Bear and the Maiden Fair||Second Sons||The Rains of Castamere||Mhysa|
|Season Four appearances|
|Two Swords||The Lion and the Rose||Breaker of Chains||Oathkeeper||First of His Name|
|The Laws of Gods and Men||Mockingbird||The Mountain and the Viper||The Watchers on the Wall||The Children|
Behind the scenes
It wasn't initially clear that the grey-haired giant seen in the Season 3 premiere was meant to be the same individual as the grey-haired giant who later appeared at the end of Season 4, despite being played by the same actor. It was later confirmed that they were the same individual in the book Inside HBO's Game of Thrones: Seasons 3 & 4.
The character has no direct counterpart in the novels. The giant never received an official name in on-screen dialogue or a designation in the credits, i.e. "Giant #3". The Inside HBO's Game of Thrones: Seasons 3 & 4 book, however, mentioned that the production team assembling the costumes took to calling him "Dongo the Doomed" by the end of Season 4 ("doomed" because he gets shot in the back with a ballista bolt). There were in fact only two specific giant costumes, for two distinct individuals: Mag the Mighty and Dongo the Doomed. For lack of any other official name, "Dongo" has been taken by Game of Thrones Wiki as the character's name.
"Mag the Mighty" is really a nickname that the human wildlings came up with for the giant whose full name in the Old Tongue is "Mag Mar Tun Doh Weg". While "Dongo" isn't a name from the novels, presumably the in-universe explanation would be that "Dongo" is a shortened nickname for a longer name in the Old Tongue (the few giant names revealed so far always have five syllables in them, written as distinct words. For example, something like "Don Goh Tun Doh Weg", etc.
In the books
In the A Song of Ice and Fire novels, Mance Rayder uses giants for breaching the tunnels at the Wall. However, none of the giants wield bows during the Battle of Castle Black. The most complex weapons that the giants seem capable of wielding are clubs and simple mauls.