Quoggoth was originally created by the many-angled chaos lord known as Shuma-Gorath and served for aeons as his disciple, but when Quoggoth began fancying himself more powerful than Shuma-Gorath, he rebelled and clashed with his former master in the skies above Earth during the time of the dinosaurs.
Though Shuma-Gorath was unable to destroy that which he had created, he was still able to defeat Quoggoth and seal him inside of a sepulcher devoted to the worship of the Old Ones which he then sunk far beneath the ocean. Centuries later, a mutant who called himself Magneto brought the sepulcher and the island upon which it sat back to the surface.
For a time, Magneto used the island as a base of operations until he was defeated by the X-Men, who ended up stranded there when Xavier's yacht broke down. While the X-Men were sitting around a campfire, an odd noise echoed through the air, Wolverine wanted to check it out but Xavier stopped him as he could not sense anything via telepathy. However, he decided to take advantage of the situation by having Kitty go with Cyclops to learn more about how recon missions work.
As the two made their way around the outside of the sepulcher, several tentacles emerged from a barnacle and tried to grab Kitty, but her screams forced them to retreat so that Cyclops could not see them. A young girl then appeared from behind one of the rune-covered pillars claiming that while she was asleep her parents went missing right after their boat crashed upon shore. Despite Cyclops' warnings, Kitty went to help the little girl only to discover that she had been a flesh puppet controlled by a massive tentacle.
After discarding the puppet, several more tentacles erupted forth from various holes in the ground and dragged both Cyclops and Kitty into the sepulcher, though Cyclops managed to free himself from their grasp, Kitty was dragged down deeper into the sepulcher where she found herself staring straight up at Quoggoth himself who told her of how he ended up trapped inside the sepulcher and then made an attempt to strike a deal with her that would ensure that all three of them escaped the sepulcher alive.
Cyclops intervened however and took Kitty with him in a mad rush to escape from Quoggoth and his demon spawn. However, the two found themselves trapped at the entrance, which not even Kitty's phasing abilities could penetrate due to the powerful binding spells Shuma-Gorath had placed on the sepulcher long ago.
Wolverine, who has on the other side of the doors, heard the two screaming at one another and used his claws to pop the crystal seal on the doors and open them, allowing Kitty and Cyclops to escape. Quoggoth managed to reach the entrance as well but before he could actually escape the sepulcher, Cyclops used his eye beams to close the doors while Wolverine put the crystal seal back in its place, trapping Quoggoth once more within the sepulcher.
Fred Van Lente created the character of Quoggoth for use in Marvel comics, and Quoggoth's first appearance is in Wolverine:First Class #12 which was released in April, 2009.
Quoggoth appears in Ultimate Marvel vs. Capcom 3 as an alternate costume for Shuma-Gorath that becomes available when you purchase Shuma-Gorath. If you purchased Shuma-Gorath for Marvel vs. Capcom 3: Fate of Two Worlds, then the data from that game's save file will unlock both him and the Quoggoth costume in Ultimate Marvel vs. Capcom 3.