Little has been revealed about Markham past. He was possibly a motorcycle stuntman or at least a motorcyclist, who at some point went insane and become a criminal. At some point, he was eventually caught by Batman and committed to an imprisonment in Arkham Asylum. After professor Achilles Milo took control of asylum, Markham become the part of his new money-making scheme. He agreed to use his talents in robbery and keep coming back to the asylum, imidietly after his job was done. Then Milo provided iron-clad alibi for him, if there was any suspicion about his criminal activity, for fifty-percent of his profits.
However one night, Batman encounters Markham during his robbery of jewelry store. During the fight, Markham's arm was broken due to unfortunate landing, but he still managed to escape and return to Arkham Asylum, thought not before his oponent managed to identify him.
Later, when Batman infiltrated asylum, he spotted injured Markham in his cell, which confirmed his suspicions about prisoner-escaping racket. Probably due to his injury, Markham didn't take part in Milo plan, to convince Batman that he is a madman who thinks he is really a caped crusader.
Madman Markham is an expentionally skillful biker and most likely an expert motrocycle stunt rider. He also use a lenghh of chain as a deadly weapon in combat. In addition to that, he's a jewelery thief.
In his first appearance, Markham is identified as Mad Dog Markham. However in his next appearance, his name was change to Madman Markham. To avoid confusion with other Mad Dog characters, the article use his second name.