The dog in question belongs to Jerry's landlady. Jerry says that his landlady is a sexual predator of sorts that is always hitting on him; however, audiences are likely to realize by even this point in the play that Jerry isn't exactly the most trustworthy of characters. Jerry tells Peter that unlike the landlady, the dog definitely does not like Jerry. Jerry says that most animals are completely indifferent about him; however, this dog chases him and tries to bite him every time he comes home.
From the very beginning he'd snarl and then go for me, to get one of my legs. Not like he was rabid, you know; he was sort of a stumbly dog, but he wasn't half-assed, either.
Jerry's solution is to try and win the dog over through kindness. Jerry figures that if he is overly nice to the dog day after day, the dog will likely stop trying to bite and chase him. Jerry provides hamburgers for the dog for multiple days in a row in an effort to get the dog to like him. That doesn't work, so Jerry goes with his original backup plan of just killing the dog.
Well, I thought about it up in my room one day, one of the times after I'd bolted upstairs, and I made up my mind. I decided: First, I'll kill the dog with kindness, and if that doesn't work . . . I'll just kill him.