Aeration of your lawn works well to help push what you have on top, down into the clay, and to break up the clay. You can rent an aerator from a rental store. If your soil it truely a "block of clay", with no sand and little organic matter in it, you need to fix that. Add sandy loam & organic matter (something like a manure top dressing), and aerate it in.

You might have a lawn expert come out and look it over - they want your business, and will do it for free. Then just do yourself what they recommend.