It should be obvious from the beginning of the play that Jerry has a difficult time establishing any kind of relationship.  From the beginning, he is aggressive.  His whole purpose is to get someone, in this case, Peter, to kill him.

Perhaps the idea of committing suicide would be an admission of failure in Jerry's mind, therefore, he must find somebody to do it for him.  Jerry needs Peter.  He needs Peter to understand why he must kill him.

Jerry's story about his relationship with the dog illustrates the idea that he is incapable of establishing  an relationship with another human being.  With the dog, Jerry is in control.

Jerry is a man who feels he has no control over his life therefore he feels that the only control he has is his death which he manipulates.