Step One: remove the old dead areas with a shovel. Trim up the sides of the area being removed so they're straight. Fill in the area with new top soil, so the new is level with the old. Now plant your seed (follow label directions). You can cover with some straw, which is good for areas that might be damaged by a heavy rain. Otherwise, just a little peat moss on top will work just fine.
When planting the seeds, don't bury them in the ground. Use just enough top soil to barely cover the seeds (no more than ¼", with less being preferable). After covering, press down on the dry soil with the back of a hoe, or use your shoe and lightly tamp it down, but don't stomp on it.
Options: There are some products that usually contain the work "patch" out there that combine a few steps and make it a little easier. They contain just about everything you need and work great if you happen to have the same grass type as the package. If not, then don't use it.
Another option is if you can find sod that matches your lawn, you can quickly repair the damaged areas. Follow all of the directions up to the point of planting seed. Only add enough topsoil so that with the sod, it matches your existing level. Sod takes about 2 weeks to get established. Keep it watered and don't let it dry out during those first 2 weeks.
Step Two: Apply a fertilizer over the area (follow label directions) that is specifically designated as a "starter fertilizer."
Step Three: Keep the soil moist. For seeds, only the top surface needs to stay moist, but (and this is important, especially if the weather turns hot) you can't let it dry out completely, particularly in the last half of the first 2 weeks after planting. Once the seeds germinate, keep the soil evenly moist and increase the amount of water, but cut back on the number of times you water. In other words, keep the soil moist at a deeper level (moist-not wet!).
Step Four: In a few weeks things will start to pop. If possible, don't walk on the areas, and don't mow the areas until the seeds are about 2" — 3" tall, about 3 weeks after the seeds first germinate. Then you can forget about it and just treat it like the rest of the area. Don't use any weed control on the new grass for at least a couple of months until it gets really established.