The easiest way is to do Samos first, Mykonos > Samos(Vathy) > Samos(Pythagorion) > Patmos.
What route you take and the order of islands when backpacking around does not matter too much. The route I have given you is pretty much an all year standard routing.
First link probably with the ferries Nissos Mykonos(3 days a week), the next link usually about 3 times a week the 'Nissos Kalimnos'(currently the Anna Express 2 times a week).
Out of season ferries tend to go for annual servicing, or get late official approval, subsidies etc. Keep flexible and there will be no problems.
The route suggested in #2 is rather long winded, expensive and needs 3 ferries. It's an option, in certain itineraries. The route from Samos down as far as Rhodes through many islands is a standard route, with cross links to the Cyclades from Samos, Kalimnos, Kos, Rhodes. Bear that in mind for ferry manouvring.