Ancestors

In what context and in what way did the Ancient Greeks worship or venerate their ancestors?

They carried out rites at the tombs of their ancestors on the anniversaries of their deaths. There's a passage quoted from a lost play of Aristophanes (Tagenistae) where a character says "We sacrifice to them with offerings, just as we do to the gods, make libations, and ask them to send good things."

At Athens there were public ceremonies. The third day of Anthesteria (Khutroi -- Pots) involved offering pots of cooked vegetables; this seems to have been for the dead in general, perhaps to take care of those who had no descendants. Another festival was Genesia, which honoured the city's war dead.