Well, ideally all of the crucial features of a standard Ruthenium or efficient organic dye, such as anchoring group, broad absorption, ground and excited state oxidation potentials well suitable to TiO2 and electrolyte for efficient electron injection and dye regeneration respectively, minimum aggregation once anchored to TiO2,  minimum recombination losses are the most important once.

I agree with those explained by Hammad. For the selection of natural dye sensitizer, you should consider that the structure has anchoring group (hydroxyl or carbonyl group), suitable HOMO - LUMO position to the conduction band or fermi level of the metal oxide you use (TiO2, ZnO, etc) and has a broad abosrption in visible regime. In our experiences, cyanidin-based natural pigments extracted from mangosteen pericarp has good performance as dye-sensitizer.

Intense absorption in visible range, profound electron turnover from incident photon, prolong excited lifetime of the electron, faster adsorption onto semiconductor surface and electron transfer via firmer grafting of dye molecules with semiconductor oxide with favorable anchorage, rapid regeneration by electrolyte, narrower gap between highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), attenuated optical energy gap, lower absorption co-efficient etc. are habitually presumed as key qualities of an ideal sensitizer for DSSCs.