Re: no grass in clay
Compost if you can afford it, if not look for horse stables or other livestock operations and get manure, which is usually all you can haul for free. The farms I've gone to would also load it.
I would rototill the area as deep as you can.. Apply the compost or manure 3" to 6" deep over the area and till it in as deep as you can. This will get you a decent growing medium to support grass. If you really want to improve the soil, go for a second 3" to 6" layer of organic material and till that in before planting grass or laying sod.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!