To fix dead spots in your lawn you should take a shovel and turn over the dirt to loosen the soil.
Then after turning over the soil where the dead spots are apply some Scotts EZ Seed Patch & Repair to the dead spots to repair the dead spots and then water the area and after a few weeks you should see some new grass growing in the dead spots.
You can also just scatter some new grass seed and fertilizer over the dead spot after turning over the soil and then water it to see new grass growing in it's spot.