Wife 'died' (she was cheating on him with Mr. Shears)
Presumably he and Mrs. Shears had a thing after their spouses were gone
He had access to the tool shed since he did the gardening
Mr. Boone killed Wellington, it's a simple indisputable fact.
Two years ago, at the same time as when it was found out that Mrs. Boone was having an affair with Mr. Shears, Chris was told that his Mother was in the hospital and died after a week. This is what I believe to be a Red Herring. But during the week that she was in the hospital, Chris wasn't allowed to visit her. We also never really see Mr. Shears during the book, meaning he probably left as well. This leads me to believe that Mr. Shears and Mrs. Boone both left their spouses together and are living together somewhere.
After they left, the spouses they left behind got together from shared grief and feelings of betrayal. My proof of this is that when Mrs. Shears came over that first time, she pulled Mr. Boone's head to her bosom. This does not often happen between neighbours unless there's something between them. And also, Mrs. Shears would come and 'sleep over' at their house, which is unusual because she has a perfectly good house just next door.
The father killed Wellington on a night where he was drunk, feeling mournful about remembering his lost spouse and the combined feeling with his special needs child. While Chris was 'detaching his mind', as Chris claims he can do which is akin to what Sherlock Holmes can do, he couldn't hear that Wellington was baying loudly and annoyingly, but Mr. Boone could hear him perfectly fine. So, in his drunken irritatedness, he grabbed the keys to Mrs. Shears' tool shed (he does her gardening, after all) and took the pitch fork from the shed, then impaled the dog.
This is the reasoning and methods behind Mr. Boone killing Wellington.
