"Oh, that's Wallace. He's teachable. I teach him to do that whenever he gets nervous."
Not much is revealed about Bill's past, although he was mentally unbalanced. To cope with the anxiety brought on by his delusions, he developed a technique involving trembling his hand. In his adult life, Bill went to college with Carla Hines and eventually entered into a relationship with her. Later, Carla gave birth to Wallace and Jesse, his sons. Carla found it difficult to raise both of the sons, so Jesse was given away and adopted by the Gentrys. Bill later left Carla under unknown circumstances (likely related to his mental health) and his mental issues began to develop even further, though he still managed to see Wallace from time to time. When Wallace started to become delusional like his father, Bill taught him everything he knew about countering his psychosis.
Bill is found by the BAU to help search for Wallace and Jesse, who both reunited under Carla's machinations and subsequently embarked on a killing spree together. However, due to his delusions (which by that point had mostly taken control of him), he barely provides any useful assistance to the team. However, Bill is brought in after one of the twins is killed by the other in a fight and the BAU is unable to identify the surviving twin. Upon observing the twin's hand twitching, he identifies him as Wallace, leading to his arrest. What happened to Bill afterwards is unknown, but he was most likely institutionalized for his mental health issues.