Christopher acknowledges the fact that his father told him not to keep thinking about who killed Wellington, but too bad: he's going to do it anyway.
He says that when people tell you to do something, it can be confusing because they often use metaphors or just aren't specific enough. (For example, instead of a sign reading "KEEP OFF THE GRASS," he would prefer a sign reading "KEEP OFF ALL THE GRASS IN THIS PARK" [59.2], so it's clear that he can walk on other areas of grass in the world.)
His teacher Siobhan understands, and always gives him very specific instructions, but no one else does. And anyway, most people break rules themselves (especially Christopher's father).
So Christopher always decides for himself what he's going to do: right now, that means finding out who killed Wellington.
That night, he goes to Mrs. Shears' house and tells her that he didn't kill Wellington: but he wants to know if she knows who did it.
She shuts the door in his face. So much for appreciating his fine detective skills.
When he's sure she isn't watching him, he goes around to her garden shed. It's locked, so he peeks in through the window, and he sees a pitchfork that looks just like the one that killed Wellington.
He figures it's unlikely that Mrs. Shears killed her own dog (considering how upset she was about the whole thing), so that leaves three options: (1) she left the shed unlocked, (2) she left the fork lying out, or (3) the dog was killed by someone who had the key to the shed.
Mrs. Shears comes outside and threatens to call the police again, so Christopher goes home. He feeds his rat Toby and feels happy about being a detective.