Do you mean to cause a mutation to occur? That's more-or-less random chance, and pretty rare. I don't think inbreeding would help, either.Question: How long, or feasible would it be to breed new color mutations? I now this process would take intense inbreeding.
Or do you mean to breed chickens that express a mutation that already happened? That can be quite fast if it's a dominant mutation, although it takes a while to get them true-breeding for the gene. A recessive gene takes longer before you have very many chickens that show it, but once you have them showing the gene they also breed true.