All breeds of chickens were developed by breeding related chickens to each other, even those that started due to mutation. Chickens that take the grand prize at major chicken shows were developed by breeding fathers to daughters or grand-daughters or sometimes brothers to sisters or other such relationships. Linebreeding is one type of inbreeding used to develop championship birds, but it is not the only model ever used.
When you breed related chickens, you reduce the genetic diversity. As Mahonri said, most people bring in new blood on a regular basis to help keep the genetic diversity up. Whether that is every year, every two years, every four years, or when they think it is time based on the chickens they see hatching varies. No hard and fast absolute rules. I think the genetic diversity is a lot more complex than just recessive-dominant genes. Flocks that don't have it seem to lose vigor over time. That does not mean it is always evident in the first generation. It could be, but then it could be several generations before you see it.
The defects that eventually show up may be physical, crossed beaks, crooked toes, something wrong with their body and fairly easy to see. It could be another trait, harder to detect. A common problem a few generations down the line is fertility of hatching eggs. They could become sickly, easy to catch diseases. If your stock has am undesirable trait that shows up in the first generation, I'd suggest getting totally new stock and starting over. There is probably a recessive gene that will be hard to get rid of.
It is not as gloomy as I am probably making it sound. If you only select to keep and breed your best stock and do not allow the ones with the traits you don't want to breed, you will probably be OK. Just bring in fresh blood every now and then. You just have to pay attention to which ones you keep.
Good luck!