I am assuming your dog is a 'show dog'. Since this is a dog show and he wins a money prize, I believe you will receive a 1099-MISC tax form in January or February, requiring you to report the amount won. However, if you do this on a regular basis and consider it a business, you can file Schedule C with your return and deduct whatever expenses you incurred for the show.
If this is a one-time occurrence, and it is only a hobby, you can deduct your expenses up to the extent of the income on Sch. C.