Origin and Characterization
The origin of Genghis Khann is shrouded in mystery. He said he was the original Genghis Khan, reborn, but there is no other information to support, rebut, or explain this.
He led an army of Huns. Both he and his men were explicitly portrayed as "metrosexual" or homosexual; their sexual identity was often made the source of jokes at their expense. He was confident and competent as a leader, however. He was presumably a good combatant himself but never fought anyone on-panel.
Genghis Khann invaded the monastery of the monks who trained the Adolescent Radioactive Black Belt Hamsters. They apparently killed or imprisoned everyone within without much difficulty.
The Dalai Momma sent a mystical message for help to the new team of Hamsters, Steven, Jean-Claude, Arnold, Lucy, and Rock, who were training in Chicago. At first they refused, but eventually decided to help. They flew to the monastery and attacked, but were quickly beaten. At least two were killed (Jean-Claude and Steven), and perhaps four. Lucy was knocked off the edge, where she fell into the hiding place of the excommunicated Master Lock and his Yeti friends. He sent Lucy off to get the old team of Hamsters together.
Meanwhile, Khann set his own plan in motion. He somehow turned some of the new team of Hamsters (Jean-Claude, Arnold, and Rock) into his accomplices. At least one of them had been killed in the earlier fight, so Khann was apparently able to resurrect him somehow. It is unknown how he gained the loyalty or control of the others--magic, brainwashing, or something else. He brought them down to the secret basement of the monastery, which held a surprise: a giant bank of computers tracking another cloud of radioactive space Jell-o, just like the one that had mutated the original Hamsters in the first place! It seemed that the monks had not been fully truthful with the Hamsters.
Khann later sent his army of Huns, led by his newly-controlled Rock, Arnold, and Jean-Claude, to attack Master Lock and his Yetis, who still lived near the monastery. They quickly beat Lock and his group, and Khann tortured him in order to get the monastery's computer passwords.
Lucy and the original Hamsters (sans Clint, who no longer felt the call of the warrior's way of life) attacked Khann and his army, but were caught. Khann showed off his control of the situation in front of them. He used the computer system to activate a homing system, calling the inhabitants of the cosmic Jell-o to come to Earth.
After gloating about his soon-to-be dominance of the Earth, he summarily shot his captives (Bruce, Jackie, Chuck, Lucy, and Master Lock) in their heads, killing them. He then welcomed the inhabitants of the cloud to Earth, saying that the monks had tried to keep them away for too long. It seems that the cloud was populated by a race of anthropomorphic hamsters, which he used as an army to take over the planet. Apparently, the Adolescent Radioactive Black Belt Hamsters were actually originally denizens of this cloud, and were jettisoned to Earth in their youth, instead of being Earth hamsters that were mutated by the cloud's energy as had been originally thought.
He ruled the Earth for ten years. Eventually Clint came to his senses and invaded Khann's compound, killing many of the hamsters in his army along the way. Khann had turned from a fit warrior to a huge couch potato. When Clint came in, he was ready to give up out of boredom. Clint wanted Khann's time machine, so he could go into the past, save his brothers, and fix his mistakes. Unfortunately, the time machine was invisible, and Clint stepped on it by accident, disappearing into the past. However, Khann still remained in his chair afterwards, implying that Clint never accomplished his mission. The last we see of him, he is watching Ameri-Khann Idol on TV, having successfully beaten both teams of Hamsters.
Abilities and Resources
It is unknown exactly what skills Khann had himself. He was presumably a good fighter, and was clearly an excellent strategist and leader.
He seemed to be very technically proficient, as he was able to master the monks' high-tech computer room and understand the truth about the cosmic Jell-o. He had a time machine, but it's unclear whether it came from the space hamsters, he built it himself, or it came from somewhere else.
He was also capable of resurrecting the dead and making people his brainwashed lackeys. Whether he did this through scientific or mystic means is unknown.