Clayface's History
Arkham Origins Incident
During the night of Christmas Eve, Basil Karlo played in an impressionist show at the Monarch Theatre. It seemed that he hadn't been turned into Clayface at that point.
Between Arkham Origins and Arkham Asylum Incident
When Karlo learned that the studio he worked for was remaking of his classic film, The Terror without his involvement, he was driven insane. Karlo then put on the costume of the film's killer Clayface and started to murder the actors and actresses in the film. Karlo's plot was eventually thwarted by Batman and Robin (Dick Grayson) when he was about to kill Bruce Wayne's fiancée (Julie Madison). Not long after, Basil managed to escape prison when his prison ambulance fell off a cliff and he tried to murder Julie once again, but was stopped by Batman once again. A couple of years later, Basil was broken out of prison by Clayface IV (Sondra Fuller). He joined her and Clayface III (Preston Payne), and they gathered the dead remains of Clayface II (Matt Hagen), and called themselves the Mud Pack. Basil tried in vain to revive Matt Hagen, but the Mud Pack had only managed to briefly revive him. Later, after the Mud Pack was defeated, Basil tricked his allies by stealing samples of their blood and managed to escape. Basil took the samples to a science lab and mixed the blood samples together and injected himself with it and became the ultimate Clayface. He then gained all of the other Clayface's powers such as Clayface III's melting power and Clayface IV's ability to gain the powers of anyone he/she was mimicking. The blood of Clayfaces II, III, and IV also give him super strength, stamina, and speed, and also made him capable of growing to an incredible size so he became a major threat to everyone who got in his way.
Road to Arkham Incident
When Clayface was committed to Arkham Asylum over the years, he was put in the Penitentiary. The doctor treating him was most likely Dr. Young, as she made personal notes about him.
Arkham Asylum Incident
Batman, while searching for Arkham's Warden, Quincy Sharp, came across a cell behind a clear wall with no bars. A sign next to it read "Danger. Do not approach. Inmate may be impersonating other personnel." The cell contained a female mannequin, white hand-prints, and excessive Clayface goo. What appeared to be security guard Aaron Cash was pacing behind the wall, and urged Batman to let him out.
ClayfaceGordon
The imprisoned Clayface impersonating James Gordon in the Security Control Room at the Penitentiary on Arkham Island
However, after rescuing the warden in the room above, another man looking like the warden appeared in the cell, claiming that the one rescued was an imposter, and begged to be let out. When Batman turned around, a morphing noise was heard and in the cell appeared what seemed to be Commissioner Gordon. He simply laughed at Batman. Detective Mode revealed that the shapeshifter had no skeleton, and scanning him revealed him as Basil Karlo/Clayface. His profile could then be read and the Riddler's riddle, "A case of mistaken identities?" was solved. Clayface commended the detective for discovering his identity, and asked him to keep it a secret, as he had a reputation to keep.
After all but one of the hidden chronicles are found around Arkham, Quincy Sharp had disappeared from his sanctuary. Clayface mentioned how the warden had left in a hurry despite Batman's insistence that he stay hidden, all the while impersonating Commissioner Gordon.
Between Arkham Asylum and Arkham City Incident
When the rescue teams of the GCPD swept through the ruined Asylum after Joker's plans, one uninformed guard believed that Sharp was trapped within the same cell that Batman saw and let him out, while in reality, it was simply Clayface taking advantage of the situation to finally escape. It was only after the real Warden Sharp was found, did the guard realize his lethal mistake, but Clayface was long gone by that point and at-large.
Basil Karlo entered Arkham City completely undetected. Assuming the guise of numerous inmates and security personnel alike, the monstrous mudman eventually made contact with the Joker, who was revealed to be his cryptic employer. At first, Karlo was reluctant to form an alliance with the dying clown, but was quickly convinced to aid the Joker and embroiled himself in the Clown Prince of Crime’s last megalomaniac master plan.
Arkham City Incident
Despite not actually being imprisoned in Arkham City, Karlo sided with the Joker. At the request of the Joker, Karlo mimicked his appearance prior to being affected by the TITAN sickness, to fool Batman, Talia al Ghul, and Hugo Strange. Karlo also agreed to move a large amount of explosives into Arkham Asylum.
With Karlo’s vast and nearly unstoppable abilities, Joker knew he had all but conquered Arkham City. Next to doubling as him, Joker used Karlo to carry out his most extreme tasks; including having the shape-shifter shut down Black Mask’s base of operations, and opening up the Sionis Steel Mill to the Joker and his gang. Karlo also impersonated Deadshot when infiltrating Penguin's operations, which resulted in Joker receiving arms directly from Penguin's armory. Joker could hide while Karlo did his bidding.
The plan was not realized by Batman until Talia was taken prisoner by the Joker at the Monarch Theatre. However, Talia managed to free herself and disarm the Joker, and proceeded to stab him. Not long afterwards did Batman deduce that Joker had been impersonated the entire time.
The actual Joker, dying from the sickness, revealed himself and shot Talia dead. Karlo arose and took on his natural form. After Batman questioned Karlo's motives for joining Joker, Karlo justified his actions, and stated that once he killed Batman, he would get the role of a lifetime (by impersonating Batman).
Batman was forced to fight Karlo by using Freeze Blasts to retrieve the cure and grab Talia's sword to slice him. As Batman froze Karlo, he sliced him into pieces. Batman and Karlo end up in the Lazarus Pit Chamber beneath the Theatre because the Joker placed explosives around the floor. Batman fought against many Clayface soldiers with his sword. Eventually, Karlo succumbed to the Freeze Blasts which Batman threw at him and then jumped into Karlo's body to retrieve the cure and sliced free from Clayface, incapacitating him. Batman then threw Talia's ceremonial sword in the Pit's controls to prevent Joker from entering the Pit. As a result, one of the generators fell down and knocked Karlo into the Lazarus Pit, creating a large explosion that destroyed most of the chamber and permanently incapacitated Karlo.
After Arkham City Incident
Karlo's dry mud was collected by GCPD when Arkham City shut down. Later, the explosives set by Clayface in Arkham Asylum were detonated, almost claiming the life of Batman who made a narrow escape.
Before Arkham Knight Incident
Samples of Karlo's mud were acquired by various parties including Simon Stagg, who created Project: Meta, and Hush which he used for his face to hide his cosmetic surgery scars. Karlo also appeared during one of Joker's fantasies, and briefly during one of Batman's simulations for Robin. Karlo is still alive within the mud collected by GCPD but unable to take form again because of the Lazarus Particles mixed with his mud and the explosion from the generator.