Don’t even think of trying to grow a lawn in the dry shade of a wooded area. Grass is already hard enough to grow in the deep shade of a building, but when it also has to deal with the root competition of a forest, it really has no chance of survival and trying to install a lawn in such a site will simply be a waste of your time and effort. The place of a lawn is in full sun or partial shade in rich soil where it has no competition whatsoever, certainly not as an understory plant where light is low and tree roots suck all the richness and moisture out of the soil.
If you want an effect of “green carpet” in the shade, think instead of shade-resistant, drought-resistant groundcovers, plants that will put up with deep shade and root competition. Go to this link for a list of suggestions: Groundcovers for Shade.