TPS, if i was doing that job i would deffinetely remove the old dead grass, or just mow it at the lowest setting. Leaving that dead grass sit their will cause an excess amount of OMATT which the soil microbes won't be able to break dwon, end result it will create thatch and a matted layer that will impede water and nutrient permeability. So deffinetely try and rake up or remove as much of the dead grass as possible. In terms of adding more soil, unless the existing soil is horrible or there are low spots in the yard that need to be leveled out, i dont see the need in putting down new soil. In terms of putting down the seed i would suggest renting a slit seeder or dethatcher with a seed attatchment that will break up the soil bed and give a nice seed layer. After you lay down the seed i would throw some starter fert in their to help with root grow, since new seedlings cant pull the phosphorus from the soil that already exist yet. Then throw down either penn mulch or straw i believe, (i get confused as to which on grows into the yard straw or hay), apply this over it so when the customer waters the yard the hay or straw and penn mulch keep the soil moist and doesn't allow it to dry out. I would then roll the area with a roller, but dont fil it up to much so it crushes everything. Fill it enough to wear it compacts it slightly to get out air pockets. You may need to come back later and spray or put down a post emergence for possible weeds that grew also. This is just my 2 cents, to each there own.