that's a tough one. There's lots of good answers about how to amend your clay soil (type clay soil into the search box in upper left of page and it will show you what all has been written here about that), but mostly they would involve ripping out your grass and starting over, which is a pain.
I don't have direct experience with it, but people say gypsum helps to break up clay soil:
"One of the biggest problems a home gardener faces in a new or established garden is a clay or hardpan type of soil. Poor drainage, soggy soil, and soil compaction are just a few of the conditions that clay or hardpan soils create.
In a new garden one can work organic humus into the soil to help break-up poor soil. Bark, sawdust, manure, compost, peat moss and soil mulches and conditioners are often used for this purpose. Incidentally, if bark or sawdust are used they will leach nitrogen from the soil, as they decompose, so additional nitrogen will need to be added on a seasonal basis.
But what do you do in an established garden? It would be too much work and take too much time to dig, replant and then recondition the soil. Here's where Gypsum may be just the answer for reconditioning the soil, because it can be applied on the surface soil in the vegetable garden, flowerbeds or on the lawn. In other words it does not have to be worked (cultivated or spaded) into the soil, it can simply be spread on the surface. What does Gypsum do? It has the ability to penetrate the millions of fine clay particles in heavy or hardpan type soils and loosen the soil structure. This process then creates air and moisture spaces that eventually loosen and break-up the soil structure. The only problem is that this doesn't happen overnight, it takes yearly applications, over about a three-year period to get the job done.
It should be noted that Gypsum does not contain any major plant nutrients, so it will still be necessary to continue a regular fertilizing program. Plus, you should continue to add organic humus in new areas, as you plant. Gypsum does contain calcium and sulfur, which are beneficial to plant growth."