Chickens' favorite thing to do is scratch and eat seeds. That is incompatible with establishing a lawn with seeds. Best time for seeding fescue or Kentucky bluegrass is September. Anything you plant now will not survive the summer heat. You could clean up sticks and leaves now, keep the weeds cut through the summer, and do whatever planting you decide to do in September. Do not till. Roundup will kill the weeds. You would do that in August. Not sure what you could do about the chickens while you're trying to get grass to grow. They would have to stay out of the area. Maybe you could get advice from other chicken owners.

Since you are in NC I highly recommend Bermuda. Withstands the heat and requires very little water after established.

Apparently late spring/early summer is the time to plant Bermuda. I have no experience with it so hopefully someone else will have some advice for you. If you have access to Bermuda sod, you can plant plugs of it and it will fill in.

With the exception of the very south east part of NC, Bermuda is VERY poor choice. The dominant and most adapted grass in most of NC is Fescue and Kentucky Bluegrass, a mixture of both grasses makes a very competitive lawn.