Alien X is still at work here, so some information may be incorrect, incomplete, or just plain missing!
Eon's Campaihg began due to the problems of the Chronian society, such as chaos, war, and a threat to their whole universe. Eon was chossen among the remaining Chronias to save them by traveling into a different universe and finding that universes version of The Hands of Armageddon and power it from the other side, in order to bring them all to a place where they could rebuild their society and create a new utopia.
Throughout his 4,000 year long journey between universes, Eon's body wore down dramatically over time however, he eventually found Ben's Multiverse and with it a solution to the problem he was facing, the Omnitrix, which could be used to create a new body for him by transforming the user into a Chronian, using himself as the DNA sample. Eon attempted to make Ben Tennyson, the user of this device, to transform many times, eventually he succeded in doing that. He repeated that process after one body reached it's limits, he would go to a different universe and obtain another Ben Tennyson as a replacement for his body and, would break the Omnitrix with no chance of ever transforming back.
Eventually after many cycles, Eon decided that he would no longer replace his body but, began his plan of obtaining unlimited power using his final body, in order to finaly save his people.