Wait until the fall or a period of the year when the clay soil is completely dry. Do not try to treat or work clay soil when it is damp as this just compacts it more.
Rent a heavy-duty garden tiller from a home improvement store or tool supply facility. Place it at the edge of the red clay and set the blade depth to 5 inches. Turn the tiller on and push it across the clay to break it up.
Place the tiller back at the starting location and increase the depth to 10 inches. Turn the tiller back on and push it over the clay again to loosen it to a total depth of 10 inches.
Spread 1 inch of compost on top of the red clay using a shovel. Leave the tiller depth at 10 inches and push it over the clay again to mix the compost into it well.
Continue adding compost 1 inch at a time and then tilling the area until you have added between 3 and 4 inches of compost.