When?

At various points the Doge of Venice was selected via acclimation by mob, propped up by whatever clique of aristocrats were most powerful at the time, selected by special committee, and eventually chosen with a complex system of electors and ballots. All these various systems had their individual pros and cons. Importantly, pretty much all elections after about the thirteenth century were driven by the need to guarantee no faction or clique would be able to exert undue influence over the Doge's selection, and thus electors were comparatively less concerned with the Doge being particularly suited for the job (although the electors did invariably settle on electing Doges with long histories of public service).

I looked at the most salient changes to the venetian government, including the Doge's selection, between the eleventh and thirteenth centuries in this older answer which you might be interested in, although I'm happy to answer follow-ups and there is always more that can be added.