Obtain moss control products from your local nursery or garden store. Look on the label for ferrous ammonium sulfate or ferrous sulfate, both of which kill moss. Apply it to your bermudagrass and the moss directly according to the package's directions.
Wait until the moss dies. The moss turns yellow, then brown as it does and becomes rough and crumbly. How fast it dies depends on the local humidity, temperature levels and intensity of shade where it grows. Rake the places where the moss grew originally. De-thatch the rest of the lawn if it has a bed of dead grass clippings or dead grass stems more than 1/2 inch thick. You can use a rake or a vertical lawnmower. You can also rent the equipment to remove the excess dead grass. Dispose of it by throwing it away or putting it in a compost pile.
Feed and water the lawn once you have removed the moss and thatch. Thatch robs the ability of water to penetrate the soil, which stresses bermudagrass.
Overseed that part of the lawn with bermudagrass seed or if the area does not receive enough sunshine, choose a grass seed that thrives in shade. If the bare spot in your yard is large, consider using sod to cover the area. When using seed, keep the area moist until the grass seed has had a chance to sprout. Then water less frequently to encourage deeper root development.