Till the top 4 to 6 inches of the clay soil. This will break up the hard clay clumps, allowing the roots of grasses to grow deeper and stronger.
Mix grass seed in with topsoil and spread this mix on the top of the tilled clay soil. By mixing the seed in with topsoil, you're adding extra nutrients added to the clay soil.
Water the new grass only enough to dampen the top 2 inches of soil. Over-watering clay soil will make it more compact.
Aerate the lawn once the grass had established itself. You can aerate the lawn with a number of different machines that punch holes into the soil. You should aerate your yard once every year just before the lawn grass growing season begins. Aerate cool weather grasses in the fall; aerate warm weather grasses in the spring.
Add a layer of topdressing once or twice every year. Topdressing is organic matter, or compost, that adds nutrients to the soil. Spread a thin layer of compost out evenly through the lawn using a wheelbarrow and shovel.