Mix a batch of equal parts topsoil and compost.
Fill the hole with the topsoil/compost mix until it is about 2 inches above the surface.
Step lightly onto the filled area to compact the mix. You don't want to pack it super-tight, you just want to press the new soil into the hole.
Apply a starter fertilizer that has a ratio of 1:1:1 to the soil.
Spread a heavy layer of seed onto the patch with your hands.
Water the seeded area daily until the seeds begin to germinate (do it twice daily if it's extremely hot). After the seeds begin to germinate, water the seeded area every other day for the next two weeks.
Push a stick into the hole to get an idea of how deep it is.
Fill the hole with gravel until it's about 3 inches beneath the surface. If the hole is shallower than that, dig it a little deeper with a trowel.
Add a topsoil/compost mix to the hole on top of the gravel until it is slightly higher than the surrounding ground.
Step on the soil lightly to compact it.
Add a starter fertilizer to the soil.
Spread a heavy layer of grass seed on top of the soil.
Water the seeded area daily until the seeds begin to germinate (do it twice daily if it's extremely hot). After the seeds begin to germinate, water the seeded area every other day for the next two weeks.