There is a product called Earth Rite and another called Mushroom Stuff.
Earth Rite actually breaks down clay soil. It breaks down clay, and increases drainage throught the soil.
I have used it a couple of times now. Once in a flower bed that was pretty much solid clay.
Follow the label direction to treat your yard and treat early spring, summer, and late fall.
Within the year, you will notice better drainage throughout the yard, increased grass growth, and gradual improvement in the texture of your soil.
In a perfect world, as noted, several cubic yards of top soil would be ideal. That isn't always an option.
With Earth Rite, you can also spray the front yard 1-2 times a year, and then anytime you notice ponding of water in the yard. Even though your sod is doing well, if there is an excessive amount of clay, it will not maintain more than a year or two.
When you seed, consider seeding as the first snow is falling. Then again in early spring, spot seeding anyplace that is coming up thin, scraggly, or bare. In the spring water Mushroom Stuff in as directed on the bottle, it will help the new grass root, and root deeper before the summer heat.
We drought terribly here during the summer. There is a drought tolerant BlueGrass mix, I believe it is either TriStar or Scott's.
I have had the best luck with Blue Grass Fescue blends for grass in hard pack and clay areas.
The other option is to add "Field Conditioner" which is essentially a Gypsum blend as discussed in earlier responses.
I would start with Earth Rite first.
Just don't spray it around any ponds. Earth Rite has been known to drain ponds by breaking down clay.
Follow up bi-annually with a weed and feed, and thin layers of organic material such as mulched leaves, manure, worm castings, cotton burr compost, etc. You don't want to bury the grass, so you may have to add thin layers a bit at a time over a couple of weeks. I use straw as well, but that is due to high traffic in my back yard (aka 260 pounds of dogs running across it packing everything down). For us, the straw gets ground in and doesn't cause an issue with thatch.
Do not mow your grass for a "final" mowing of the season late in the year.
Let it get long enough to go to seed before winter sets in. It will seed itself that way. We allow our back yard to go to seed every late fall/winter. It helps to increase density AND it is reseeding with the healthiest strains of grass that grow well in your yard.
A year or two? You'll forget you ever struggled with a patchy, scraggly lawn!