i faced the same problem 10 yrs ago.
i started by applying 3 in 1 lawn feed weed and mosskiller. after applying I waited for 3 weeks then raked out all the dead moss. at this point i wished i hadnt started as there was hardly any grass left at all, it was like a bomb site. i bought some new grass seed and scattered it over the whole lawn then sprinkled a light covering of peat based compost over the seed and rolled it down. after about 3 weeks the grass came through and by the end of the summer the lawn looked fab. trouble is it's a lot of hard work every yr as the moss comes back, have to feed, scarify, aerate, top dress. unfortunately my back's not up to it anymore so i'm trying to find a local gardener who can do it for me.