William was in fact, not mentally unstable like the rest of variants encountered in the asylum as stated by Pauline Glick 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. While Tiffany was conversing with agents Paul Marion and Pauline Glick, she mentioned that Billy was in fact, not mentally ill, but that she had sold him to Murkoff for a large sum of money. After hearing this, Billy attacked Tiffany, telling her that he "loved her." Billy then tore her apart while the two agents ecaped. The caravan itself was then destroyed by an explosion, presumably ending Billy's life for good.
Billy is depicted as being a well-built man with blue eyes and no hair, in his early 20s. The Morphogenic Engine has had very little effect on him, although his skin is covered with bulging veins and other scarring. Hope is shirtless, donning only a pair of blue underwear and handcuffs around his wrists. Whilst being strapped in one of the Engine's pods, a number of tubes can be seen entering every orifice in his body.