Obtain sterile, well sifted compost which you can either buy in bulk or as bags from a garden center or home supply store. According to the Garden Counselor website, you will need 1 cubic foot of compost for every 50 square feet of lawn. Make sure the bag is labeled "sifted," since unsifted compost contains large chunks which will prevent the grass seed from germinating.
Spread the compost over the lawn using a shovel to sprinkle it on in a very thin, 1/4-inch layer. Do not throw large clumps of the compost over the soil and then attempt to spread it around; this will move the underlying grass seed around and result in an uneven growth pattern.
Water the compost with a 1/4 inch of water form a garden hose just until the top of it is moist. This is only to prevent it from blowing away if high winds occur during the storm. Do not over-saturate the area with water.