Have you checked out the info provided by the extension service (NC State University)? They have an excellent lawn care calendar that is even available as a smart phone or tablet app.

There is really minimal difference in scheduling or calendering a lawn care regime under an organic program or a non-organic program........it is merely a matter of the products and practices utilized. Timing remains the same.

In my personal opinion, I think you are going to be starting out with difficulties based simply on the lack of a well-prepared soil base. It makes no difference how a lawn is planted - seeded or sodded. Both methods require the same basic preparation of a fertile, fast draining and adequately deep soil bed. I would encourage you to address aeration and addition of organic matter to the lawn on a frequent and ongoing basis to achieve that end. It is very much a cart before the horse situation now but with proper attention and time it can be corrected.

Here is a link that might be useful: NCSU lawn care calender

To grow a good healthy lawn requires a good healthy soil and to get that means having a good reliable soil test, for soil pH and major nutrients, done. For North Carolina residents this soil test is often without charge.
In addition, these simple so9il tests may be of some use,
1) Soil test for organic matter. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. For example, a good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top.

2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drainsâ too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer your soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy.

Once you know more about the soil you have then you can male plans to make it into a good healthy soil that will grow a strong and healthy turf.

Here is a link that might be useful: NCSU CES