A cost effective chemical for moss and some broad leaf weeds is sulphate of iron. I would initially spray this in solution over the mossed areas at this time of year, and then follow the advice of Thunderbird+.
In the spring when the ground has warmed up, spray the whole lawn with the same, as it will also burn the leaves of broad leaved weeds when thay are active, as well as killing any new moss.
The spring treatment should be followed by the treatment suggested by Thunderbird+, especially due to the fact that bare areas left by the removal of dead moss will allow weeds to flourish if left.