You may think clay soil is a poor soil, but that is not entirely true. Clay soil holds a lot of good nutrients because the clay absorbs and holds on to them. The problem with clay is mainly the drainage issue. It doesn't drain well at all and causes standing water in low spots that creates potential mold problems. In other areas, water tends to just run off, not giving the grass a chance to absorb it.
Sow a grass seed, such as Bermuda grass, that is versatile and will grow in clay soil. If you don't have the time or funds to amend your soil, this is the easiest way to go.
Test the soil's pH level and adjust to a level that is good for grass growth by adding amendments to the lawn (see Resources). The pH level should be 6.0 or higher for healthy grass.
Aerate the lawn and spread the seed. The aeration will help break up the soil a bit and give the seeds somewhere to take hold.
Water the lawn to start the seeds' germination, and cover with straw so the grass seed won't wash away.
Till the soil up if you want to have a healthier grass. Till sand into the soil as well as some organic material. Use a rake to smooth it out. It won't be perfect soil, but it will drain better. Clay soil that drains is always an improvement
Test the soil pH balance and add amendments, if needed, to get the pH above 6.0 (see Resources).
Spread whatever grass seed you choose, making sure that it is at least tolerant to clay and your weather conditions.
Cover with straw and water until damp.