Prior to Crisis on Infinite On Earth
The first Kryptonite Man was originally Kryptonite Kid. He was a criminal, alien teenage. He flew through a green kryptonite cloud. This gave him the proprieties of Kryptonite. He battled Superman.
The second Kryptonite Man was the ruler of Krypton prior Superman's race existing. His race had put themselves into stasis. When Krypton exploded, he was the only survivor. He gained the ability to eat kryptonite radiation and gain super strength and durability when he did so. However, he became dependent on it. After stealing a ship from aliens called Seeders, he went to Earth to kill Superman (thinking Superman's race destroyed Krypton). However, he was stopped by angry Seeders.
Post-Crisis on Infinite on Earth and Prior to The New 52
Spies Simyan and Mokkari created a clone of Superman. The clone was called Kryptonite Man despite having no Kryptonite-related powers. This version had all of Superman's powers.
In Superman / Batman, an energy being, of the same name, was created in battle between Captain Atom and Major Force. This being had the power to generate Kryptonite radiation, possess people like a demon and heal its hosts.
Scientist K. Russell Abernathy was doing a Kryptonite-related experiment. When the experiment exploded, the radiation to fused to Abernathy, thus leading to him become the fifth Kryptonite Man. As part to test his theories, he tried to attack Kryptonians. He was defeated and imprisoned. Lex Luthor helped him break out of prison. Later, he attacked Superman, but was stopped by Krypto and Jimmy Olsen.
The New 52
Clay Ramsay was a wife-beater that was thrown into a bay by Superman. Wanting revenge, Clay joined "Project K-Man". Thanks to the project, he gained the ability to see radiation spectrums and fire Kryptonite eye beams. However, a side effect was him losing his ability to be rational and became a berserker maniac. Naturally, he became a super villain that fought Superman.