Like a lot of things in the Batman universe (such as Batman and Robin themselves), Clayface began as Basil Karlo, an actor who used the identity of a character he portrayed in a movie to murder the cast of a remake of the film and commit other crimes. He made his first appearance in Detective Comics in 1940.
During the latter part of the 1950s, Bat-villains started becomine more sci-fi in nature, and adventurer Matt Hagen gained shape-shifting powers after becoming exposed to radioactive material. He was the new Clayface until the 1970s, when Preston Payne used Hagan's DNA to try and make a cure for an ailment he suffered from to become the third Clayface.
The fourth Clayface, a woman named Sondra Fuller, fell in love with Payne and the two had a child together, Cassius, who became the fifth Clayface (get it? Cassius Clayface?), taking on the powers of both his parents.
He was followed by Dr. Peter Malley, who could not only shape-shift, but could melt people without touching them.
There have been eight Clayfaces in all, and aside from Karlo, all have had shape-shifting powers. Some have had super-strength and the ability to divide themselves into multiple parts. All have been enemies of Batman.