Horrible joke, if you could even call it that.
The thing that really stood out in the article to me, was the players referring to themselves as a family and wanting to take care of business for the head man in the family.
It implies that Kiffin is so dumb that he would sell a talking dog just b/c it lies to him. That's all I've got as to why it might be offensive.
My preacher actually told the same joke (sans Kiffin) a few months ago, and his jokes are notoriously bad. Him and West must buy the same joke book.