It seems pretty clear that at least some level of belief in the gods was part of community life among the ancient Greeks, just as it was for the Romans. Note that community life is the important point, not personal faith. There was a multitude of gods and and goddesses in the polytheistic Mediterranean world; in the Greek world, each polis had a particular patron deity. The god might be the same as the neighboring polis' patron deity, but cultic observances might be different, or each polis might worship a different aspect of the same god. Greeks invoked gods in sacrifices that were part and parcel of civil life and their civil -- sacred and secular were meshed -- festivals. Leaders sought the gods' "opinions", if that is the right word, through some form of divination before any important undertaking. People wore amulets to ward off evil spirits. Some joined mystery cults. Writers wrote stories with conflicting details about divine-human interaction. Important families proudly traced their ancestry to the gods -- or the sons of gods, the legendary heroes who populate their myths.
Festivals -- like the dramatic festivals in which the great Greek tragedians competed and the ancient panhellenic games, like the Olympics -- were held to honor the gods, as well as to tie the community together. Sacrifices meant communities shared a meal, not only with their fellow citizens, but with the gods. Proper observances meant the gods were more likely to look kindly on the mortals and help them.
Yet there was some awareness that there were natural explanations for natural phenomena otherwise attributed to the pleasure or displeasure of the deities. Some philosophers and poets criticized the supernatural focus of the prevailing polytheism:
Socrates was charged with failing to believe properly and paid for his unpatriotic religious belief with his life.
Homer and Hesiod have attributed to the gods
all sorts of things which are matters of reproach and censure among men:
theft, adultery and mutual deceit. (frag. 11)
But if horses or oxen or lions had hands
or could draw with their hands and accomplish such works as men,
horses would draw the figures of the gods as similar to horses, and the oxen as similar to oxen,
and they would make the bodies
of the sort which each of them had. (frag. 15)
"Socrates is guilty of crime in refusing to recognise the gods acknowledged by the state, and importing strange divinities of his own; he is further guilty of corrupting the young."
From Xenophanes. See What Was the Charge Against Socrates?
We can't read their minds, but we can make speculative statements. Perhaps the ancient Greeks extrapolated from their observations and powers of reasoning -- something they mastered and passed down to us -- to construct an allegorical world view. In his book on the subject, Did the Greeks Believe Their Myths?, Paul Veyne writes:
"Myth is truthful, but figuratively so. It is not historical truth mixed with lies; it is a high philosophical teaching that is entirely true, on the condition that, instead of taking it literally, one sees in it an allegory."
Also see: "Greek Religion" from Oxford History of Greece in www.centenary.edu/religion/dotto/rel320/oxford%20article.pdf