What I did with my red clay yard that was rock hard and wouldn't perk. I found a horse stable about 5 miles from where I live that had a large pile of well rotted stable cleanings - free for the taking. I hauled enough stable cleanings (called 'black gold') to cover the yard at least 6 inches deep. I then wet it down until it was wet to the clay. I waited a week and went to roto-tilling it into the clay. When I had it all cut into the top 6 inches of clay, I spread 200 pounds of high nitrogen general purpose fertilizer per 1000 sq. ft. and another 6 inches of "black gold" and roto-tilled it into the soil. The roto-tiller I used could go to 12 inches deep, so when I finished the second roto-tilling I had 12 inches of loam that would crumble between your fingers and held water like a sponge.
That was 6 years ago and my lawn is still dense, green and never needs watering, even here in GA where we had been in a severe drought for 4 years until lately. I do add lime and fertilizer each year per soil test recommendations.