First of all, you need to carefully choose the grass variety. There are some varieties that are more tolerant of the shade and there are some which cannot survive without the sun. Fine Fescue St. Augustin are one of those which can tolerate shade. Discuss with the local nursery and choose carefully. With grass grown in shady areas, you might have to use more seeds than in normal areas to get a thicker resistant turf.
Trim the branches of your tree to improve sunlight. You can trim off excess branches without harming the health and shape of the plants.
When mowing the grass in the shady area, leave the height a bit longer than the ones in the sunlight. The grass needs to have more surface area to trap whatever sunlight it can get. When you mow, remove the clippings from the top immediately so that the air and light supply to the grass is not blocked.
Water the grass infrequently but when you do, drain well. Frequent but light watering leads to weaker roots.
Grass grown in shady area requires almost half as much nitrogen as that required by normal grass. When you fertilize, keep the nitrogen level lower to keep diseases away.
Do not let plant leaves and clippings cover the grass for long, especially in autumn. They will create more shade for the grass and increase the problems. Shade area grass also needs a greater amount of fungicide and pest control because the humidity and lower temperature in the shade encourages diseases. You will also have to put extra effort on removing the weeds and shrubs which love the shade.