Ku'Gath was once a Nurgling s(h)itting on Nurgle’s shoulder. Whilst Nurgle was mixing his most virulent disease ever, Ku'Gath did a barrel roll, and tumbled from his seat place into the cauldron, where he took a great draught of the lovely contents and swelled with its power. Invigorated, Ku'Gath drank and drank until the cauldron was empty (the fat shit) and he had grown into a Great Unclean One. Nurgle laughed at the antics of his Nurgling, who had become the embodiment of the perfect disease that had been bubbling within the cauldron. Ku'Gath, however, realised that he had robbed his father of the greatest disease ever concocted, and has since been wandering the universe, searching for the perfect ingredients to recreate the toxin for Nurgle. He also captures the guys he infects and take them back to his lair, where he can observe their degeneration more closely.
He met Mortarion when the Primarch came upon him crying, in despair, in the Garden of Nurgle. Upon asking why he was sad, Ku'Gath told him about how he became a Great Unclean One. After they talked for a bit, Ku'Gath told Mortarion where he could find his foster father's soul.
At some point, Ku'Gath was summoned on the Agri-World of Curwen at the head of the Daemonic Legion. A Grey Knights strike force arrived, and Anval Thawn led a force into the very heart of the Daemonic horde to prevent being surrounded. Ku'Gath swallowed the dude whole, but Thawn detonated his weapon inside the Daemon's throat and burst from its body, and unsurprisingly, Ku'Gath was slain and banished back to the Warp. Ku’Gath appears again in M41, when intervention of Styer and his Grey Knights in the attempts of Inquisitor Malia Orbiana to find biological weapons against Orks (which long story short, the Orks were immune to), which led to the manifestation of the Bonewrack disease, Ku’Gath’s once lost infection, in realspace. The Mary Sues managed to defeat Ku’Gath and his horde of Plaguebearers and expel daemons to the warp, while Ku’Gath himself was wrecked by the blow of Librarian Gared when his Nemesis Greatsword bisected Ku'Gath's head. He later led Nurgle's forces in the Invasion of Ultramar during the Plague Wars.
On the TT
Ku’gath will wreck you, no joke. BUT to use him you are going to have to not only buy imperial Armor, but either the Obsolete Finecast Model, or the expensive as fuck Forge World one. Probably best to use a normal Great Unclean One and say that the fat fuck decided to walk for once. He also costs a ton of points.
Strangely enough, Ku'gath failed to make the transition from 7th edition to 8th, and his place as "named Greater Daemon of Nurgle" was instead replaced by Rotigus, who hasn't had any presence until this edition.