I don't know where this "Inbreeding is bad for show chickens" came from. Inbreeding is how all chicken breed were initially developed. Inbreeding is how show chicken lines are developed so they can win prizes. Inbreeding is how show chicken lines are maintained so they can continue to win prize ribbons, but after the championship line is created they use certain techniques to maintain some genetic diversity. This mainly involves breeding cousins instead of parent/offspring or siblings such as the spiral breeding technique.
@Sungrove Farmstead you do not get a blank check for inbreeding however. There can be some effects. You need to use your observation skills and reasoning ability. But inbreeding isn't as bad as some people fear.
A standard model used for thousands of years on small farms was to keep replacement hens and roosters from your own flock for several chicken generations. How many generations? That depended on what they saw and the basic genetic make-up of their flock. The more inbreeding (or lack of genetic diversity, same thing) the higher the possibility of problems showing up. These problems could be physical deformity, poor health, poor productivity, poor fertility, stuff like that. The more chickens involved each generation the more diverse the genetics, so the more generations you can usually go before problems develop. If you keep a flock of one rooster and four hens you could possibly see problems in just a few generations. If you keep a flock of 20 roosters and 200 hens like some hatcheries do you could pretty much continue indefinitely.
Part of this working well is dependent on you making good decisions on which chickens get to breed. If you see chickens that have traits you don't want then don't breed them. This could be chickens that have physical deformities, grow slowly and poorly, or that don't lay well. Do you want hens that go broody? If you have one that goes broody and you like broody hens, hatch her eggs. If you don't like broody hens, don't hatch her eggs. We all have different traits we like.
This is getting long so I'll sum up. If you are getting a decent hatch rate, the chicks you are hatching are healthy, and you are OK with their productivity you are OK. If you see declines bring in a fresh rooster and start over. You can keep the same hens.