Ouch.
Sounds like you need an expensive, multi-stage remediation.
You'll first need to plow under your entire yard, then mix in 4%-6% clean fine sand, based on the volume of clay assuming a three-foot deep box. Ideally, you'd re-till, then wait for 10 days or until the next soaking rain. At that point, you want to remove 30% - 40% of the volume of the new mixture (assuming you want to keep the same, level yard), then replace it with organic humus. (Essentially, you've just built the ideal loam.)
A truly standup guy would introduce a colony of earthworms, grubs and other beneficial insects to add life (ha!) and longevity to your soil.
Finally add a healthy coating of grass seed, daily watering - unless it rains - and hopefully lots of sunshine. You'll want to keep humans and large animals off the grass while it's growing, so a temporary fence (or webbing) may be necessary depending on your setup.
In no time, you'll have ultimate bragging rights and will put your neighbors lawns to shame.
