yep the black stuff is the dead moss--- you have to scarifie this out and dump it- the lawn may look a bit sick after a good scarifieing -- then do the weed an feed bit --if you are concerned about the horrible looking lawn you will now have --top up the lawn with a sprinkiling of grass seed-- but dont cut with a lawn mower for a few weeks to allow the new seed to get established the you should only cut about a third of the new growth off (this allows the new seed to harden up and form stronger roots) and then over a period of time you can lower the cut on the mower to the desired length
the moss will come back -- but to try and slow down the rate at which it returns a good spiking with a garden fork over the entire lawn to aid drainage will help followed by filling these holes with kiln dried sand and then a good sprinkling over the surface of the lawn will help to curtail the new growth of moss
the moss is probably caused by overhanging trees(lawn shadowed by trees?) poor drianage or a heavy thatch from not picking up grass cuttings
Use a cylinder cut mower instead of a rotary, and as monstermob's said, only take 1/3 of the length off each cut. It's a pain but worth it, during the summer months 2 cuts a week is normal if you want a lush tidy lawn.