A surveyers transit and a laser level are the best way to get the slope right. But if that sounds like too much trouble or cost. Practice with a level and string or laser pointer to establish a flat line and put stakes down with inches marked to establish eight inches downward or whatever depth is needed. You don't want a callback or do over this--you dont want them to call you and say, "This is wrong; water is pooling in the southeast corner!"
If the red clay soil will grow weeds--you can grow grass. I would add lime if the soil turns out to be acid below 6. Get the customer to pay for a professional test and give him the results.