As soon as Genesis was born, everyone knew it was a terrible thing.
God himself fled Heaven, leaving the angels in charge. The Adelphi were given the job of keeping Genesis caged, but they soon discovered that this was a nearly impossible task - Genesis was a being of almost infinite power, unless it had a host to guide it.
So Genesis escaped, and fled to Earth to find its host - Jesse Custer, the reverend of a small parish in Annville, Texas. Why Genesis chose Custer is unknown, but the process of possessing the young preacher released an explosion that destroyed the church and nearly everyone in it. Custer was the only survivor. He emerged without a scratch, and with a new and frightening ability: The Voice of Command. Whatever he hold a person to do using the Voice, they would do it. No matter what, as he found out to his dismay when he told Sheriff Root, "Go fuck yourself."
Genesis was, in effect, hiding inside Jesse's head. The angels wanted it back, and had sent the Saint Of Killers to get it. The Grail, a worldwide group promoting Armageddon, knew about Genesis and wanted to use Jesse for their ends. Genesis, for its part, didn't seem to want to be found. Because it knew things. Things that could set in motion tragedy after tragedy, and once Custer found those things out, tragedy was in good supply.
One of the things that Custer discovered, or figured out, was that God was a raving egomaniac, who fed off the love of other beings. In guiding the angel and the demon to their tryst, God had created a being as powerful as He was, and all He wanted was its love.
In the end, that need for love undid God, but not before Genesis was forced out of Jesse Custer's body. Where Genesis is now, in a Godless universe, is unknown.
Although a similar character appeared in Hellblazer, (an offspring of an Angel and a Demon), Garth Ennis has explained that the two characters are similar but not the same. This was first found in issue #28 of Preacher. In the back of the book (the questions to the writer section), Garth clearly states
"Don't let a dealer make you pay extra for a back issue of Hellblazer cause it contains the origin of Preacher's Genesis...cause it don't"