Looks an interesting walk. I haven't done it in one go, but I have done all the individual bits as parts of other walks.
It looks as though for the sections over Bleaklow and Kinder you can follow the Pennine Way. From Edale to Chapel you can walk up to Rushup Edge and then follow paths to the South of the road (marked on the map) into the town.
Chapel to Cat and Fiddle is a bit more problematic, as there are lots of different possible routes. I would suggest making your way over to the Erewood Resevoirs and following the valley up to the pub on the top.
Cat and Fiddle to Buxton is straightforward. Reverse the part of the route you came up from Erewood Resevoir to somewhere near Cheshire Bridge and follow the path down to Burbage on the edge of Buxton.
Hope that helps.