At least four superheroes have fought comic book crime under this name, and one villain has used the identity to try and cover his tracks.
The first, a cross between Batman and Tarzan, patrolled the Golden Age. His adventures appeared in Crash Comics Adventures #4 and 5 in 1940, and in the 32 issues of Cat-man Comics (Fiction House Comics).
Like Edgar Rice Burroughs' Jungle Lord, young David Merrywether found himself orphaned in the wilds-- in his case, of Burma-- after bandits killed the rest of his family. Wild tigers raised him. This should have left him feral and severely disadvantaged, but instead, he developed various feline abilities-- excellent night-vision and agility-- and, for no apparent reason, superhuman strength. Initially, he had nine lives, and could return from apparent death. This aspect was soon dropped; possibly, its limited number of uses suggested a lack of faith in the character's longevity.
He adventured briefly in the jungle-- cat-cowled, for some reason-- before finding his way to America. Having lost his parents to crime, he naturally took a dim view of criminal activity. He first tried his hand as a private eye, and also underwent military training. Finally, he became a superhero, using an outfit that recalled more than a little the one worn by DC Comics' Batman, colored red and yellow. A case for a lawsuit, like the one launched against Fawcett Comics over Captain Marvel's resemblance to Superman, could have been made.
He even took on a sidekick, a girl named Katie Conn, who called herself the Kitten, and wore a variation of Merrywether's costume. In the tradition of Captain America's Bucky, she somehow has the ability to keep pace with her metahuman mentor, despite an absence of special abilities. Unlike most youthful sidekicks of the era, the Kitten grew up, and over the course of the 1940s the girl blossomed into a vivacious young woman, prompting the comic to become a source of ribald commentary as to the actual nature of the heroes' relationship.
In the 1990s, these comments were validated by AC Comics, who posited a "Vault of Heroes" project during which various 1940s characters to whom they'd gained the rights were supposedly placed in suspended animation, to be revived when needed. In AC's retcon of comic history, Conn had been older when she first met Merrywether, and they had married on VJ Day. They were brought into the Vault of Heroes some time after the war.
In the 1980s, a Cat-man appeared in Australia, resembling the original. He has a similar origin, a familar (though predominantly green) outfit, and a Kitten sidekick. He is engaged to another woman, however, whose father, conveniently, is a scientist who aids the Cat-man when needed.
In the 1970s, an entirely unrelated character appeared on the pages of Captain Canuck. This unhyphenated Catman was Jason Corey, who disappeared into the Canadian wilderness while hunting the legendary sasquatch. He was rescued by a stranded extraterrestrial who, in fact, communed with the sasquatch community. The alien did not want Corey to leave, however, because he feared his existence would be exposed to the world. Nevertheless, the cunning human made away, taking with him an alien device which gives him superhuman strength and immunity to cold temperatures (remember, this is a Canadian superhero). He, too, fights crime, wearing a purple and orange outfit that gives no indication of why Corey might have chosen a feline totem.
Batman, of course, has experienced a lengthy love-hate relationship with a Catwoman, but he also has encountered a Cat-man/Catman who resembles uncannily the Fiction House hero who in turn had been a swipe of the Dark Knight. His name is Tom Blake. Originally, he was a hunter who turned to crime. His costume was yellow and red, and he shared Selina Kyle's obsession with cat-themed crimes and gizmos. He later became a more sinister character with a mystically-powered African cloak and a less cheery costume. In the early twenty-first century, writers revised Tom Blake yet again into more of an anti-hero than an outright villain.
This Tom Blake Catman was the second-rate villian of past comics, whose lack of success had made him a figure of contempt and ridicule. He returned to Africa to regain his pride. His skills improved, he now hunts DC heroes and villains when he believes they have abused their power.
In the early 1990s, a Catman appeared in two comic-book-like camp chop socky flicks, Lethal Track and Boxer's Blow. This Catman is a secret agent who gains kung-fu-enhancing superhuman abilities after being scratched by a radioactive cat.
Finally, Peter Criss of 70s supergroup Kiss called himself "Cat-man" in Kiss Meets the Phantom of the Park, in which film he fought crime using various cat-like abilities. He also appeared in Marvel's Kiss Comics, but was there known as "Cat."
AC Comics Online. http://accomics.com/
International Catalogue of Superheroes. http://www.internationalhero.co.uk/gacatman.htm
Steve Rogers. The Golden Years Library. http://home.insightbb.com/~GoldenYears/Library.html
Mark Cotta Vaz. Tales of the Dark Knight: Batman's First Fifty Years. New York: Ballentine, 1989.