When Murdock had found Billy during a walk he'd taken when Face had conned one of the doctors into giving him a 48 hour pass, he'd just been a tiny puppy which had been nuzzling against his dead mother. The rest of the litter were already dead, and it was only Billy that was left. Billy's mother had to have been the largest and ugliest dog he'd ever seen. He didn't know what breed the mostly furless thing which had to have started decaying already based on its appearance was, but Billy looked to be part Terrier. He was certainly small enough to be one.
He'd picked Billy up, named him Billy since it was as good a name as any, and tucked him in his jacket before going to a nearby store and buying baby formula and an eye dropper with the money that Face had given him when he'd set out for his little stroll in order to give Face a bit of privacy. After sitting on a park bench for a while, and getting the puppy to drink the formula from the eye dropper, he headed back, reasonably certain that his friend had been alone with his companion long enough. Face hadn't noticed Billy when he'd returned to the apartment his friend had scammed, but then again Face had a tendency to be rather self-absorbed. When he'd asked Face to look after Billy until he got sprung again for a mission or whatever, Face had said no before dumping him and Billy off back at the V.A..
He waited for several hours before finally going in because he needed a bed for the night because he knew that the hospital staff would confiscate his puppy the instant they realized he had it. Tucking Billy back into his jacket, he did his best to smuggle the small puppy which had already been through so much inside. Through some miracle, he had managed to make it back to his room without anybody spotting his new pet.
In the beginning, every day that he cared for Billy who had been much too young to have been orphaned like that, he worried that Billy would be found and taken from him. Eventually, a week had passed, and he became worried for a different reason. None of the hospital staff had seen Billy on the day he had carried the tiny puppy everywhere with him because it had been looking sick, and he had been afraid that it would die if it were left alone. In fact, one of them had tried to humor him when he'd mentioned that he was carrying a sick puppy on his person.
Ever since he could remember, he had been able to see things that other people swore weren't there, and able to know things he shouldn't know. His grandma had told him that the things he saw were real, and that it had been one of those things that people said weren't real that had killed his Ma who had been out hunting them like she did since after something had done something very bad to his father before he was born. Things could have gone worse for him and his unit in Vietnam if he hadn't been able to instinctively sense when someone or something wasn't what it appeared.
Billy obviously wasn't an ordinary dog, but the question then became, what was he, and how dangerous would he be when he got bigger? That, and was the dog really there, or just another symptom of his environment and the medication he'd been placed on? He'd seen a couple of things that weren't there weren't there during his time at the VA, and he hoped that Billy wasn't one of them. When nobody else could see what you could see, it tended to erode your faith in reality.
It had been a pair of Hunters who had been intent on killing Billy while he'd been taking him for a walk when he was six months old who had eased a few of his fears, and clued him in to exactly what Billy was. As he'd seen in the time he'd had him, Billy wasn't dangerous like the two men had claimed he was. He was keeping him, and if anyone wanted to kill Billy, they would have to go through him first.
Besides, Billy was more dog than Hellhound, and they were perfectly suited for each-other if the story his grandma had told him about his father coming home with black eyes nine months before he was born was true.