There are several ways to get rid of grass lawn, and none of them are easy. The best organic treatment I know of is to layer the yard with newspapers, wet it down thoroughly, cover with landscape cloth, weight the edges down with rocks or bricks, and let it sit for an entire season. The newpapers will decompose, and attract earthworms, which will help spread the nutrients from the decomposing grass and paper throughout the soil. It’s a ton of work.
Personally, I would get a tiller, till the whole thing up, pull out any rocks or other debris, rake out most of the sod, and then till it again, down about a foot. Then you add plenty of compost, till that in, and also sand, to help with drainage, if needed. It is usually a good idea to test your soil first, to see whether you have acid soil or alkaline soil, and then add the proper stuff like bone meal or whatever to get the perfect pH.
My soil is heavy clay with high acid content, and it has been a battle to get it just right. All my gardening is organic. I’ve never used a harsh chemical on any part of my yard.
Another thing you might want to do is to keep the leaves that fall from your trees and the grass clippings from the rest of the yard to use as mulch. I’d check a library for a good book on the organic process, as it is not for the weak willed. It is never easy, but nothing good ever is.