There are dozens of articles on the web about how to seed a lawn.
First though, pick a seed that is 'the best' you can afford. It pays dividends in the long run.
There are shade seeds but no grass can grow where the shade is deep; there must be some sun.
One way to improve the sun coverage is to remove the lower branches of trees that shade their base. If you can prune back some of the upper branches that block sunlight, that too can help grow grass underneath.
You mention tree roots are coming up all over the lawn...but speak nothing about what trees you have.
Tree roots can be cut back according to a formulae that takes into account the health and stability of the tree.
Do not cover over these roots with more than 3" of soil.
Putting more than that can cause suffocation of the roots and can cause untold harm to the tree itself.
Many articles point this fact out.
The formulae I speak of is by measuring the trunk of the tree up 3 1/2'...there, take the measure of the diameter, multiply that by 3 and that figure is the number of feet out from the base you can safely cut roots.
Thus if the measure is 6" diameter, it would be 18 feet out from the base of the trunk that roots can be safely cut.
Minor roots can be cut anytime...they replenish themselves quickly but don't cut major roots without going out far enough so that the stability is not undermined.

Try to till in as much as you can without cutting itno the roots at greater than safe depths. Compost and other organic material can help retain moisture in your clay soil which has the problem of not releasing what stored moisture it has.
If the problem of the clay soil is felt to be too much, read material on the use of 'gypsum'. This material can be 'an amendment, conditioner, and fertilizer.
It amends soil alkalinity, lowers high pH, raises low pH conditons and leaches out harmful salts.
It can condition and improve soil structure allowing for air and water penetration and root development. It nourishes the soil with calcium and sulfur.
But read the many articles describing what it does for lawns and soil.

If straw is used to top the overseedng be sure it is "old" straw...not fresh, which has seeds and if so used they drop and cause your lawn to become a barnyard.
Watering a seeded lawn is done by sprinkling often...3, 4 times a day to begin with, for about a week then cutting back the amount and not as often the next week, then gradually reducing that to normal levels as the grass needs it.

You might wish to use a starter fertilizer with the seeding which can improve the germination of the seed.

The one thing that can improve your lawn, beside using the best seed, is to improve the level of sunlight that reaches the lawn. After it comes in another help that can help you attain a lush lawn is to mow at heights that help to shade out weeds. Two and half to three inches height is recommended for most grasses except some of the southern types.

A picture would be worth a thousand words here. Bringing in soil might be good if you suspect the original soil has been washed away for some reason. Some trees will react to hard soil by sending their roots out at the surface. Then as the tree ages, the roots become larger and expose themselves above the surface. If that is what has happened, then adding soil on top of the natural grade will cause you drainage problems and could cost you extensively if water pours into your basement (worst case). Adding soil should be done with considerable forethought as to the consequences. Another thought on adding soil is to keep it lower than the root bark on the trees. Tree roots flare out from the base of the trunk. If you bury the root flare completely, then you are burying tree bark. Tree bark will often rot when it is below the soil level. It is best if you leave some root flare above the soil level to be sure you are not burying tree bark.

Tilling is a bad idea for many reasons. One of the obvious ones is that you have roots everywhere. You can't till tree roots. A much less obvious reason is that tilling destroys soil structure by fluffing it up. Once it is fluffy it has to settle again. That process takes 3 years. When it finally settles it will settle unevenly leaving your surface bumpy.

Grass seed should never be mixed in with surface soil. It always goes on top of the surface. Many people recommend "scratching" it into the surface but that is not necessary. Nor is covering it with straw. All you need to do is loosen the soil at the surface (1/16 inch deep) before you apply the seed, then apply the seed, and finally roll the seed down with a water filled roller (rented). Seed should remain at the surface. If you insist on straw, as goren said, be careful not to use hay. Hay is forage for animals and has seed in it. Straw is the left over after the seed is threshed out.

New seed must be watered 3x per day at breakfast, lunch, and dinner for about 10 minutes. The idea is to provide enough consistent moisture for the seed to swell and open (germinate). Unfortunately these are the exact conditions that crabgrass seed needs. For this reason grass seeding is most successfully done in the fall after crabgrass has stopped germinating. Also grass sprouted now will have weak roots that cannot stand up to the summer heat and dry weather. That might not be that much of an issue in your area.

Don't assume you have clay. Fill a mayonnaise jar half full of dirt. Then fill the jar completely with water and a drop of soap. Swish it around and let it settle. You will see rocks and sand settling immediately to the bottom of the jar. In an hour all the silt will have settled. At that time if you have clay, the water will be cloudy. Let it sit another day and see if the water clears up. If it clears then you do not have clay. Clay will cloud the water for days and days. There are several conditions of minerals that cause soil to seem to be clay when in fact there is no clay at all. Don't be too surprised by this. Adding gypsum without doing this test and a soil chemistry test can be a mistake.

I disagree with goren on mowing height. The only grasses that need to be mowed lower than 3 inches are bermuda, centipede, bentgrass, and some zoysias. Kentucky bluegrass (which you can't grow in shade) should be no lower than 3 inches. Fescues and all the other grass types should be mowed back to 4 inches. Usually this is a mower's highest setting. Grass grown in the shade needs to be longer simply so it can photosynthesize.

Can you post pictures showing the tree roots and one from a distance showing the trees with the canopy?

Thanks for your responses. dchall, I am doing the soil test you suggested to see about the clay content. I have photos below showing the mostly dirt area with roots running across it and then the tree itself. Hopefully, they are helpful. And after looking at the area again, it's not really shady. Maybe a little shady in the morning, but plenty of afternoon/evening sun. To restate my goals again; I would like to get more grass growing in the area again and then do something in order to avoid running over the roots with the lawn tractor constantly. Thanks for any help.
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