I've laid the odd lawn or two in my time from seed so just to back the points up then.
Clay soil - no problem. Grass is after all a weed and will grow practically anywhere.
Drilling is essential to guarantee germination (or use a fork is you have plenty of time on your hands). This shouldn't be sparsely done either. Grass seed won't germinate, especially on clay, if left lying on the top of the soil.
Nitrogen. Agreed. A young grass crop mated with nitrogen in the soil produces a lush growth. Perhaps the cheapest form of nitrogen can be applied by using sulphate of ammonia, a basic chemical compound. Although farmers may have more readily available, cheaper sources of chemicals ey Stik?
Organic? Well, blood and bone mix is more slow release. Chicken manure is good if only because it adds organic matter to your clay soil. I have used neither of latter btw.
Just wondering what mix do they use at military fields to get the ground so smooth? Grass at Syerston is like a bowling green.