1. Apply alfalfa pellets (yes, rabbit food) at a rate of 20 pounds per 1,000 square feet three weeks prior to your seeding date.
2. Scalp the existing weeds to the ground with a rented mower (you don't want to ruin your own mower).
3. Seed heavily with your favorite seed (and you really do get what you pay for in seed). The bag will tell you how much to seed. Don't skimp.
4. Roll that seed down with a rented, water fillable, roller. This will put the seed into intimate contact with the soil and help the seed stay moist between watering.
5. Water twice a day for 10-15 minutes for two weeks until the seed germinates and gets up.
6. Reevaluate your germination at the two weeks point. If you don't have the grass density you want in places, reseed right away.
7. Set your mower at the highest setting and mow when the grass is tall enough to cut at that height.
8. After you have the grass density you want, start to back off on the watering. Skip a day and water for 20 minutes. Do that a couple times and skip to every 4 days for 40 minutes. Where you're headed is weekly watering in the heat of summer and less frequently during the rest of the year.
If you completely cover the seed with peat moss or something else, you will lose some of the seed. Grass seed, unlike other plant seeds, germinate at the surface, not below the surface. In the great plains of the world, the seed is knocked down by the grazing animals and pressed relatively gently into the surface of the soil with their hooves.
In the spring you can add preemergent if you want to. Apply Corn Gluten Meal when the forsythia is in bloom. Besides being a organic pre-emergent, it is also about 10% nitrogen. Minimum to apply is 20lb per 1000 sq ft, twice a year.
If you are going to try an organic program, fertilize about 3 weeks before your last freeze date.