I don't know the categories open to you, what the judging criteria is, or how good your judge is. This will be different for a lot of us. That makes it hard to get too specific on recommendations.
If you are just looking for a production breed, it may not matter that much. With a 4-H group, there should be a category that fits abut any chicken. But if you have a pretty well qualified judge that does understand the standards, your best bet is to not get a hatchery chicken but search for a decent breeder in your area, someone that breeds birds for show. You can search for Buckeyes, Chanteclers or Wyandottes, those with the small combs, but in reality any of the dual purpose breeds should suit your purposes. I'm leaving the EE's off because there are no breed standards for them and Ameraucanas are not considered real good layers. They are all winter hardy, you just have to watch out for frostbite. You'll probably be showing them before frostbite is a concern anyway.
Since it is 4-H, talk to your county extension agent. They should be involved in 4-H to start with and they probably know some breeders in your area. Many chicken people are willing to help out 4-Hers.
Good luck and enjoy!