Clay soil is made up of very fine particles. These particles cling tightly together when they get wet and become dense and hard like bricks when they dry. It is very hard for roots to penetrate into the surface of clay soil, and the microorganisms that normally keep soil healthy can not live below the surface, since there is no oxygen in the dense soil to nourish them. As a result, wet clay lawns require extra care to stay healthy.
One of the biggest problems with wet clay is that it does not drain well. Excessive water will pool on the surface and damage the roots or, if you have a steep grade, will simply run off into the storm gutter and be completely wasted. Water your lawn with only half an inch of water about three times a week. This will give your grass enough to drink without drowning the shallow roots.
Clay soil is, by definition, compact. It lacks the holes that topsoil has to let water drain through. Aerate your lawn twice a year with a plug core aerator to help improve drainage, decrease thatch and increase root depth. You should also take care to not do things which further compact your lawn. Keep foot traffic to a minimum, since people walking over your soil will tend to compress it more.
Over time, nature creates a layer of topsoil out of decaying organic matter. You can speed up the process by how you care for your clay lawn. Store good, loamy topsoil in an area where it will stay dry. Every couple weeks, place some in a broadcast spreader and apply a very thin layer over your clay lawn. Over time, this will build up into a layer of topsoil gradually enough that it won't destroy your lawn. You should also use a mulching lawnmower to add more organic material back into your lawn every time you mow.