I agree, the lawn is mostly dead or weeds.
I would use roundup and kill everything.Only use regular Round Up or glyphosate with no other additives for extended control especially to plant after. You will need to water after applying the herbicide and it can take three weeks or more for some of the weeds to yellow and die. Remove the dead plants and water again. If more weed seeds come up repeat Round Up until you don't see weeds come up for at least three weeks with regular watering.
After you remove the dead grass and weeds, it should not be composted but trashed.
Rent a tiller. Find out where your utilities are and how deep. Most should be deep enough but you want to be sure you won't rip anything out. If there is a sprinkler system in there, you have to know the location of that. Rototil 4-6 inches deep, if you see roots or rocks, you will have to mark it or move it by hand. Rocks can become missiles or break the tines of a tiller. Large roots will have to be dug out if they are in the way. If you see any weed seeds or nuts from sedge remove it by hand.
Cover the ground with 4-6 inches of a good blended compost and 1/2 inch of manure over the soil and work it into the soil until it is well blended with the soil.
This is important. Grade the soil so that water will flow away from the foundation. If you have a swale make sure it is not filled in. If you don't have a swale you can always make one. Swales and rain gardens direct water out of the yard away from the foundation or a rain garden directs water to a place to temporarily store it to prevent runoff from polluting the streams it runs into
https://yardcare.toro.com/create/plantin ... -problems/
If you have a large lawn and you don't have a couple of hours a week to hand water, now is the time to install a sprinkler system. It can be a do it yourself thing, but if you don't like to dig trenches and are not good with working with pipes, valves and timers, this might be something you might want to hire out.
Zoysia is a perfect grass for me. ( I like emerald because it can crowd out weeds and it grows slow), but I live in Hawaii and zoysia is a warm season grass. You said you live in the northeastern US. Zoysia may not be the best choice for you. You can google best turf grass for your state and add the .edu extention for a recommendation for grasses suitable for your area.
https://www.angieslist.com/articles/cho ... theast.htm
Publications from extension services are available specific for your locality like this one for Massachusetts.
https://ag.umass.edu/resources/home-law ... eets/lawns
You may be able to seed a lawn. Others would have to advise you on this. Very few grasses are seeded in Hawaii. Most lawns are grown from plugs or stolons. Grass that grows from seed here are too high maintenance.
What ever grass you choose, learn how to care for it and get it growing healthy. A healthy lawn is thick enough to compete with many weeds and mowed at the right height it can discourage others.
There are too many weeds and the condition is so poor, it will be better to replace than overseed.
Happy gardening in Hawaii. Gardens are where people grow.