Through history, there has been various different Kryptonite men,with different origins but all them sharing the same quality of being a Superman enemy capable of emit a deadly (green) radiation of Kryptonite than could kill Superman.
The first Kryptonite Man was really Kryptonite Kid. The Pre-Crisis version of the Kryptonite Man appeared in only a few stories. He was a teenage alien criminal who flew through a cloud of Kryptonite, turning him into a super-powered being. He fought Superboy as the Kryptonite Kid. This version was used in the out of continuity story "Whatever Happened to the Man of Tomorrow?" in which he is recruited by Brainiac for a final battle against Superman. He is killed by Krypto, who tears his throat out before succumbing to Kryptonite poisoning.
The second Kryptonite Man was the last survivor from a prehistoric race from Krypton. As being native of Krypton he shared many of the abilities associated to Superman, plus the radiation powers derived from Kryptonite exposition.
The Evil Factory
The first post-Crisis version as an irradiated, green skinned Superman clone created by Simyan and Mokkari. Essentially, he was a Superman clone with green skin. He was given the name Number Two. He ended up trapped in a colorless dimension with his creators.
A second Post-Crisis version was a sentient cloud of Kryptonite radiation that originated when Captain Atom absorbed the radiation of an incoming Kryptonite meteor. The radiation was able to hop from body to body after Hiro Okamura siphoned it from Atom. He would posses at different times the bodies of Captain Atom, Batman, Monsters and became temporally part of the Composite Man.
One year Later
A more later version post crisis is the Dr K. Russell Abernathy, a brilliant Metropolis researcher determined to use the unique radioactivity found in the pieces of Superman's home world for useful purposes. His early experiments used monkeys as test subjects, but Abernathy's growing boldness led to kryptonite infecting every cell of his body. Driven insane by the radioactive poisoning, he went on a rampage until stopped by Supergirl, who had taken over Superman's duties while the Man of Steel regained his powers following the Infinite Crisis. She defeats him by using her heat vision to melt a tanker truck around him, shielding her from his radiation and binding him until the police arrest him.
Incarcerated for this crime, the Kryptonite Man became the prisoner of Lex Luthor, who used him as a power source during his hunt for a buried Kryptonian battleship, using his concentrated radiation to power the long dormant ship.. The Kryptonite Man escaped prison a second time with the help from his irradiated lab monkey. His plans to obtain funding from underworld kingpins came to an end when Jimmy Olsen burst into his lair under the identity of "Mister Action". The Kryptonite Man gained the upper hand until the arrival of Krypto the Superdog.
After Flashpoint another version of the character debuted. This time he was Clay Ramsay, a wife beater captured by Superman. He volunteering to Project K-Man, a project run by Lex Luthor.
Ramsay eventually will join the Anti-Superman Army along with other people with kryptonite related powers: the Blue Kryptonite Man and the Red Kryptonite Woman. The trio is know as the K-men.