Use a shovel to dig up and remove the dead sod. For mostly bare spots, use a metal garden rake to remove the remaining dead grass and loosen the soil.
Pour 1 part grass seed and 3 parts soil into a 5-gallon bucket. Use a garden trowel to mix the 2 together.
Pour the soil/seed mix onto the bare patches. Use the metal rake to level and smooth the surface. Tighten a garden-hose nozzle onto the end of your garden hose and set it to a light spray. Water the patches until they are soaked, but not muddy.
Water the patches 2 to 3 times per day. If the patch dries out, the grass seed will die. Continue with this water regimen for about 3 weeks.