Well I'm a little confused by your terminology but I'll give it a shot.

First of all, repeat three times slowly, with a deep breath in between: There's Nothing Wrong With Clay. Clay Is Good.

Clay holds water and nutrients. That's a good thing. You won't have to water or fertilize so often. I wish I had clay. I have sandy gravelly till, and lawns do real poorly in that. No clay to speak of, so it dries out faster than I care to water it, and the sand erodes away in between rocks and clumps of grass, so it's all hillocky and rocky and hard to walk on.

No, chipping doug firs to cover clay soil will not be a good idea. Your grass would be growing in wood chips - not a good idea.

I'm not sure what you mean by top soil, if it's been covered by the clay. Top soil is what's on top. If you have distinct layers of clay and other stuff like loam or sand, then you need to till to blend the layers together.

You will want to sift out rocks and larger gravel pieces if you have a lot of that. Clay usually doesn't have a lot of rock in it, and that's good. Lawns don't like rocks.

Who told you you don't want dirt??? that's what plants grow in. Plants don't do well in pure sand, or pure compost, or pure mulch, or pure bark. They need dirt. You may need to add compost and till it in, but it needs to be mixed in to the 'dirt'. 'Dirt', aka soil, is a blend of materials: clay, sand, silt, compost.

Then you want a meadow mix probably, rather than a lawn seed. Fescues do well here, along with bentgrass. Ryes are ok, but you don't want a lot of bluegrass - it's too dry here in summer and too wet in winter. And you don't want pure rye as it doesn't knit together into a turf by itself. Develops holes and bare spots real fast. You can get 'ecoturf' and 'erosion control turf' and 'meadow mix' which will be blends of easy-care grasses with flowers such as white clover (an excellent choice), alyssum, baby blue eyes etc. I've also used pasture mix, which is a blend of taller coarser grasses intended for grazing animals, and usually red clover. That's the kind of thing you wade through rather than walk on.

Then in fall you buy bulbs of early-blooming crocus and plant those in the lawn, along with galanthus and chionodoxa. Spectacular in late winter! And if you can get the native early blue violet, Viola adunca, or the evergreen violet, Viola sempervirens, those go well in lawns too. Nonnative violas do well in lawns too: Viola odorata and Viola labradorica.

Native meadow grasses, if you can find them, are Festuca rubra, Festuca idahoensis, Festuca roemeri, Danthonia californica, Carex inops.

I would contour the lawn so as to give it a good crown. You can then top dress with a sandy mix topsoil prior to planting. Puddles in a lawn is a no no, especially clay soil.