The fallen leaves from Walnut trees are toxic to other plants. The leaves should not be added to a compost pile. You will need to rake up the leaves quickly when they fall onto the grass. Assuming your tree creates a good deal of shade, you could try growing a fine fescue. They do well in cool conditions and shade.
There are several types of fine fescues. Depending on where you live, plains or mountains, you can investigate which type might be worth a try in your location. It is difficult to keep grass growing under shade trees and keep the tree healthy at the same time. They will have different watering needs and depending on the age of the tree, grass & added soil could interfere with the roots of the tree and its abilitiy to absorb oxygen.
You can consider planting around the outside of the dripline of the tree rather than directly underneath. With nurturing, you may be able to have success.