William was in fact, not mentally unstable like the rest of variants encountered in the asylum as stated by Pauline before he was being sent into the Morphogenic Engine and he was actually sold by his mother to Murkoff for a large sum of money.
William Hope was the first host of the Walrider, and would be his only physically restrained host. He might believe that Rudolf Wernicke is his father, as he's kept him alive through the powers of the Walrider. He can be seen at the end of the game in the Underground Lab. He's in a sphere, presumably glass, with many tubes and instruments running into and out of his body. He's being kept alive by a large reservoir, filled with various chemicals, and is constantly being kept in a lucid dream state, in order to work as efficiently as possible with the Walrider. The large machine hosting the entire process, the Morphogenic Engine, is present feet away from his sphere. After Miles kills Billy by stopping the flow of fluids from the reservoir and turns off power to his machine, he bleeds out inside the machine rapidly. The Walrider then leaves Billy's corpse, and Miles becomes the new host.
Despite being apparently killed by Miles, the Walrider, possessing Billy's body, later turns up at the caravan of Billy's mother, Tiffany Hope. The body of Billy was then, after Tiffany was slaughtered by the Walrider, blown up in an explosion, presumably ending Billy's life for good.