The clay soil prevents drainage and is easily compacted. This prevents grass from sending out roots and establishing itself. The weeds are hardier and require less perfect conditions. You need to break up the soil and possibly add some amendments (sand, organics), then reseed.
When you rake up the debris you can throw that in a compost heap as is.
The rototiller will break up your soil and any remaining grass, leaving a field of aerated dirt, perfect for adding ammendments and reseeding. If there are only patches, you should rototill the patches and reseed but also aerate your yard. You can have people aerate for a cost, if you have a tractor you can purchase/rent a plug aerator. The biggest reason for bare spots is compacted soil. Then use a weed/feed and you should be okay.