Trim the tree branches less than 8 feet high with a chainsaw to allow more sunlight and rain to filter through to the ground.
Cover the areas where you're trying to grow the grass with a mixture of compost and topsoil in a layer 2 to 4 inches deep.
Set a rototiller to a depth of 6 inches and run over the topsoil/compost mix to mix it into the subsoil.
Fill a lawn roller one-third of the way full with water, and run it over the topsoil.
Add a starter fertilizer with a ratio of 1:1:1 or 1:2:1 to the topsoil. Add enough water to make the topsoil damp. Avoid getting any fertilizer on the healthy areas of your lawn as it can burn the grass. The ratio indicates how much nitrogen, phosphorous and potassium is in the fertilizer.
Spread a heavy layer of grass seed onto the topsoil. If the area you're seeding is large, use a seed spreader and make two passes, one going side-to-side and the other going up-and-down. For smaller areas, spread the seed with your hands. Consult your local home and garden center for advice on which grass works best in your area, but fescues are usually the grass of choice -- they grow extremely well in dry, shady conditions.
Cover the seeds with a 1/2-inch layer of topsoil to protect them from being washed away by the rain or getting eaten by birds.
Water the seeded area with a fine, misty spray. Do this every other day until the seeds begin to germinate plus two weeks afterward, then add a day between waterings. Keep people off the grass while it is growing.
Raise the deck of your lawn mower so it's cutting the grass in the shaded area an inch higher than the sunny areas of your lawn.
Water the grass weekly, providing 1 inch of water. If you've received little or no rain that week, add enough water to get the soil damp to a depth of 6 inches (this can take 2 to 3 hours).