Answer: Fescue lawn seed is the shade grass seed you need. Overseed spring and fall. Adding more dirt continuously will add to the mess and takes time, but it can’t hurt. (Top Dressing is the type of dirt you need) Fertilize often, maybe 4-5 times as trees are absorbing nutrients. Cut your grass long at 3 inches. Water entire area a few times a week in the summer months. This should improve things a little.
Alternative:
Or: Don’t bother growing a lawn. If you live in an older established neighborhood with lots of trees, trying to grow any kind of lawn in that environment is very difficult.
My solution is to replace as much of the dead sparse lawn as you can. Why fight it when you could slowly establish a yard you could be happy with. Circle tree stumps with wall blocks 2 or 3 feet from the stump one or two high. Fill area with dirt and plant things that will grow in the shade and need little water. Periwinkle, Pachysandra and Hostas do well under most trees. They will also do great in other parts of the yard. Euonymus plants can cover a shaded corner area nicely. Plant a few Yew bushes in another area. They may not grow big, but they will survive.
Planting walking stones in all areas that you have a lot of foot traffic, or just areas that you walk to. You can never have a large enough patio where no grass is going to survive. Don’t fight a shaded yard, slowly work to replace it.
Best plan: Depending on your budget, artificial lawns look incredible in heavily treed yards.