The soil should actually be fine (plenty of us grow grasses in clay soils). But you can certainly remove the top six inches of soil in that area and replace with a good quality top soil if you wish.

Don't remove too much less than that if you replace the soil as the roots dislike moving from one soil type to another. They'll tend to do what they would in a clay pot--stop at the interface, which will lead to short-rooted grass that never looks great.

Other than that, seed, and you can gently rake the surface to loosen it up if you want. It's already been heavily disturbed, so the normal rules about not tilling don't apply, although I'd still avoid it.

On a slant, I'd be inclined to put burlap down to reduce erosion while the grass grows. Tack it down at least at the edges well enough that flowing water won't easily go under the burlap and lift it.

Water gently two to three times a day until it sprouts fully (about a month if the mix contains bluegrass, two weeks if it's fescue and/or ryegrass), then back off slowly on the watering to allow the grass to adjust.

If you're concerned about the health of that clay soil, once the grass is in, you can scatter some compost over it. That will help speed up the process of getting beneficial microbes into the surface material. If you want to go the extra mile, also add some organic fertilizer at the same time as the compost (or before or after).