The first step is to tidy up the existing grass. Remove thatch with a garden or dethatching rake, and remove weeds, with herbicide or a weeding tool, depending on your gardening philosophy. If you pull enough dandelions, you can probably skip the next step: aerate the lawn. A pitchfork, or even a garden rake, can do an adequate job, but an aerating tool is best for extremely compacted lawns. Aerating will improve the health of the grass, and make watering more effective, as compacted soil creates more run-off. If you find any bare patches, loosen the soil with a cultivator in that place. If your lawn has a lot of perennial grassy weeds (such as quackgrass), you may be better off starting from scratch after killing the weeds. Use an herbicide or soil sterilization to kill everything in that area, then plant into the dead vegetation.
The next step is to add a top-dressing. Two 3.8 cubic foot bales of peat moss mixed with a 110 litre bag of vermiculite will create a light, weed-free cover for 1000 square feet of lawn that will improve moisture retention. A 1/2 inch thick top-dressing of aged manure or compost can be applied instead, which will add both nutrients and organic matter to the lawn soil.
Spread two pounds of grass seed per 1000 square feet. Be sure that the seed comes in contact with the soil by lightly raking it, or make it safe from birds by sprinkling soil or compost over it. Use a seed mixture that uses Rye, Fescue and Kentucky bluegrass. Avoid cheap mixes that use high percentages of annual grasses that will look good the first year, but won't persist, and may even contain undesirable weed seeds.
If desired, apply a slow-release fertilizer high in nitrogen (such as 24-4-8) according to package directions. Water, and keep the lawn moist until the seed has germinated, about three weeks (for the Kentucky bluegrass, less for other varieties). Seeding is best done in the fall; hot summer days will quickly kill the new grass, and although early spring is also a good time, there will be more competition from annual weeds.
Try to stay off the lawn as much as possible until the new grass is well established. For the best results, reseed every couple of years, or as needed. Maintain your new grass by not mowing it shorter than three inches, and giving it one inch of water (about one hour of watering) per week. A healthy lawn is less work in the long run, so avoid costly replacement by doing routine maintenance.