Appearance and powers
As the title would suggest, Leviathan is the ruling deity over the human urges of lust and gluttony, in also seems to have control over human cruelty and extreme fear. Under its dominion the cenobites practice an extreme version of sadomasochism.
The abstract entity knon as Leviathan appears under the form of a massive, silver diamond floating in the center of cenobite's home realm, The Labyrinth. From its center, the cenobite deity is emmiting several black beams of light which force those who cross their path to experiance past memories.
Hellbound: Hellraiser 2
A recently resurectedJulia Cotton claimed to had been given freedome from Leviathan's world in order to bring Doctor Phillip Channard inside the god's realm. Channard was brought by Julia infront of Leviathan that used it's black light o reveal the doctor's disturbing youth, identifying him as a sociopath. Julia then reveals to the doctor that Leviathan had always intended to use her as a means of bringing more souls from Earth to The Labyrinth in order to creat more followers. A distraught Channard is pushed by Julia inside a massive chamber in which he is transformed into a Cenobite. Moments later, Channard emmerges from the chamber, regretting his former reluctance. To further the transformation Leviathan connects with the newly appointed cenobite by inserting a phallic shaped tentacle inside the doctor's head. Together, the two travel to Earth where they begin capturing the deranged patiences of the Channard Mental Institution, Channard himself having left these people several Lament Configurations to solve.
Ironically, it was the the Lament Configuration which would prove to be Leviathan's undoing. During Kirsty's visit inside The Labyrinth, Pinhead and his entourage transform the puzzle box into a diamond, rendering it useless. One of Channard's patience, a seemingly semi-catatonic young girl by the name of Tiffany, happens to be a master at puzzle solving. Seeing the psichological horrors of The Labyrinth and terrified by the Cenobite Channard, she is motiveted into solving the puzzle box, once again. With ease Tiffany is able to reshuffle the diamond back into it's initial form, this somehow shifts Leviathan into a larger scale puzzle box which resembles the Lament Configuration, causing the souls of countless victims to flee in a rapid rate and returning back to Earth. Channard is decapitated in the process.
Whether this killed Leviathan is unknown; however, he has not appeared since and in future films Pinhead himself becomes the one in charge of the Lament Configuration, trapping the souls of its victims not in the Labyrinth, but in their own hell.
Hellraiser VIII: Hellword
In this installemnt of the Hellraiser movie series the myths and rumors surrounding the cenobites have become the bassis for a massive multiplayer online role playing game, Hellworld. A large mansion in which most of the movie's plot takes place is named The Leviathan House, in honor of the god. It is revealed that this mansion had been, infact, designed by non other then Philip Lemerchand himself, creator of the Lament Configurations. The movie's climax reveals that cenobites do exit, which would also imply that so did Leviathan.
BOOM Comics, Hellraiser series
According to the BOOM comic books, Leviathan is obsessed with law and order, and views humans as chaotic beings. Because of this, it uses Cenobites as it's foot soldiers against chaos and flesh.