With the pickax, break up the hard, clay soil to a depth of at least one foot. Use the pointy end of the pickax to break the soil and the other end to slice up clumps of clay.
Rake the entire area, further breaking up any clumps with the edge of the rake or your hands.
Mix in topsoil with the existing soil and blend thoroughly. The mixture should be at least one-half topsoil; you might need to cart off some of the excess soil you dug out with the pickax. If you like, you may add peat moss or mulch, as well.
Apply water liberally to further break down and soften any remaining clumps of clay soil. Mix further with your rake.
Let stand overnight, then apply gypsum to the surface using a lawn fertilizer spreader. Use about 40 pounds of gypsum for every 1,000 square feet of dirt. The gypsum will penetrate any remaining fine clay particles and loosen the soil structure, creating air and moisture spaces critical for root growth and development. Water thoroughly and immediately.