Clay soil is some of the worst soil for trying to grow things. The soil's characteristics, including the tendency to compact, make it difficult for grass seed to take root, become established and grow. However, the condition of the soil can be improved by breaking up the soil through a process called aeration. Aeration allows air to get down into the soil and improves the texture of soil. Soil will also loosen if it has been amended with humus or other compost.
Rent a soil aerating machine from a garden center.
Use the soil aerating machine to aerate your lawn. A soil aerating machine aerates your lawn by removing plugs of soil from the lawn. The machine is operated similar to a lawn mower. The motor starts with a pull-rope, and the settings can be configured to determine how deeply the machine will aerate the soil. Once the machine has been started, run it over the lawn in rows, exactly the way you would run a lawn mower.
Spread finished compost over the lawn with a shovel. Rake the compost with a garden rake into an even layer that is 1-inch thick.
Work the compost into the grass using a leaf rake.
Turn on a sprinkler and water the lawn deeply (there should be 1-inch of standing water on the lawn). This will help work the compost into the holes left by the aeration machine. As the compost decomposes, it will help to loosen up the clay soil.