There are two versions of Overman. The first comes from a world where all superheroes were part of a genetic experiment.
The second Overman, is Karl Kant, a.k.a. Kal-L, whose rocket from Krypton crashed in a field in Czechoslovakia in 1938.
Grim n' gritty Overman
The first Overman was from a "grim-n-gritty" world (pre-crisis earth-17) where he was the first superhero, and all the others were grown from his cells. This version of the Justice League was attached to the military. Overman was infected with an STD, causing him to go insane and killing and (it is implied) eating the other heroes. This world was presumably destroyed during the Crisis on Infinite Earths. He appeared first in Animal Man, as part of the second crisis.
Psycho-Pirate was recreating characters from the multiverse, when he felt Overman trying to come out. Pirate realized how horrible this was and did his best to contain him, but Overman was recreated anyways.
He appeared carrying a giant bomb with the words "Purification Day" written on it. Overman lurches around Arkham Asylum, carrying the bomb and drooling like a madman. Various other characters including Ultraman, Bizarro, and even Streaky try to stop him, but it's not until Animal Man arrives and travels between panels that Overman is caught in an ever-shrinking comics panel and destroyed.
Overman appears in Infinite Crisis as one of a number of Superman analogues that Alex Luthor attempts to merge together.
Nazi Overman (Earth-10)
Rocketed to Earth as a baby from the dying world of Krypton, Kal-El landed in the occupied Sudetenland in 1938. From the rocket that carried him, Nazi scientists reverse-engineered the technology to win the war, and Overman became the hero of "a utopia built on human suffering" . He became affiliated with the JL-Axis; consisting of Brunhilde, Leatherwing, and Underwaterman, while fighting Uncle Sam and His Freedom Fighters.
However, his love of his duties was tempered by the guilt he carried. Eventually, Nazi scientists succeeded in seeding Overman's genetic material with two human children. One of them became a faliure, a monster called Antihuman; the second one however was a success and became the heroine called Overgirl. Overman became very fond of Overgirl and came to regard her as his cousin.
A blonde variation of Overman appeared during Countdown to Adventure. His characteristics physical and behavior differ vastly from what was stablished by Morrison later with Overman: this blonde Overman was a racist facist and viollent and is probable than his existance is considered not canon now.
However his companions of the JLAxis appeared in a dream secuence in Mastermen.
During the Final Crisis, Overgirl went missing. In the course of looking for her, he was contacted by Monitor Zillo Valla with an offer - if he helped her find a way to save the universe, she would help him find his cousin. He accepted her terms.
On the adventure that followed , he helped save several worlds from the crashing form of the Echo of Midnight. When Zillo Valla's craft crashed on Earth-51, she asked him to take her to her quarters. However, this was a ruse, and she used his blood to revitalise herself and her craft.
Overman survived and attempted to attack Valla, but was stopped by Billy Batson, who told him that Valla had acted only to save the ship. Shortly after, he joined his fellow Supermen in attacking Mandrakk's destroyer.
While he survived the adventure, he was unable to find out what happened to his cousin as Valla was killed before being able to inform him of her fate. Overman ultimately learned of his cousin's fate from The Question.
Tortured by the guilt of what the nazis made during WW2 and the following years, Overman has tried to built an utopia, triying to forget the sins over what is built his world. However is possible than his own selfdoubt will weak his determination to fight against Uncle Sam and his Freedom Fighters, who want to see their world burn.