how can i grow grass in a backyard that is mostly clay?
Clay soil has good and bad features. The soil particles are extremely tiny--much smaller than loam or sand. It takes longer for water to filter in between the particles, so it absorbs water more slowly. A slow gentle rain works well, but short cloudbursts can result in water running off and not being absorbed.
On the other hand, once it gets wet, it holds onto water longer than other soil types.
In order to improve it, add organic matter. Clay soil plus organic matter = top soil. Ideally, organic matter is tilled in before the lawn is seeded. However, you can still incorporate organic matter into your soil.
1. Grass cycle: when you mow, let the grass clipping remain (unless they are so thick they'd smother the grass). The clippings will filter down between the grass blades and decompose into the soil. Worms, ants and other soil critters will mix it down further in the soil profile.
2. Topdress: Spread a layer of organic amendments (dry composted material or manures) each year in the fall. About 1/2" should be fine. Aerating and then topdressing is also a good practice as this allows the organics to fill in the aeration holes.
3. Mow leaves and let those particles decompose into the turf. This can only be done to a point. When leaves get thick on the ground, they will need to be removed altogether.
Read through this helpful guide for more ideas about successful seeding: http://extension.umd.edu/sites/default/files/_images/programs/hgic/Publications/HG102%20Lawn%20Estab...