Some Greek philosophers like Xenophanes detested the humanized gods found in Homer and Hesiod for their deplorable actions and observed humanity's tendencies to fashion (false) gods in their own image. Xenophanes instead proposed the existence of one god "who is like mortals in neither body nor mind."

"Man made his gods, and furnished them with his own body, voice and garments.

If a horse or lion or a slow ox had I go hands for painting and sculpture, the horse would make his God a horse, the ox would sculpt an ox.

Our gods have flat noses and black skins say the Ethiopians. The Thracians say our gods have red hair and hazel eyes." - Xenophanes

Others like Theogenes viewed the mythology as allegory and by no means a literal narrative.

Edit: direct quote

Source: Classical Mythology: Images and Insights

Greek Lyric Poetry, trans. Willis Barnstone

Did Sokrates believe in the standard Greek gods? Not really. If you asked him, he would have said "yes". But this is because natural philosophers of the time routinely took their ideas about the elemental forces at work in the cosmos and equated these forces with the traditional gods: it was a kind of interpretatio philosophica, if you will. It equated "Zeus" with the rule of order and will; "Night" with the primordial chaos that underlies the tangible universe; "Kronos" with time (chronos, by wordplay); and so on. This interpretatio philosophica followed in the footsteps of sixth century BCE allegorists who had begun interpreting the gods as representing tangible physical forces (e.g. "Poseidon" = water, "Hephaistos" = fire, etc.), but among fifth-century thinkers the natural forces in question became more and more abstract.

As a sample, here are a couple of quotations from the Derveni papyrus, a late fifth-century tract that presents an allegorical interpretation of a traditional mythological poem attributed to Orpheus. Column 17 of the papyrus:

For "Fog" existed before the solidification of the present universe, and will always exist; that is to say, it did not come into being, but (always) existed. The reason why it got called "Fog" was outlined above. It was imagined that it came into being, because it was named "Zeus" -- as if previously it did not exist. And (Orpheus) says that it will be "last", because it was named "Zeus" and this name will persist in existing for it until the present universe solidifies into the same form in which it was previously suspended. (Orpheus) shows that this is why the universe became the way it now is, and then again became in this...(he) indicates this in this line:

Zeus is head, Zeus is centre, all things were wrought from Zeus.

And column 18:

When (Orpheus) says "Fate", he refers to how this earth and all the rest of the universe is suspended in the Fog, which is a current. And Orpheus gives this current the name "Fate". Other people use expressions like "Fate spun for them", and "all these things will be as Fate spun"; and they speak rightly, but without knowing what "Fate" or "spinning" is. For Orpheus used the name "Fate" to refer to Will. To him, this seemed the most suitable of the names that all people gave it. For before Fate was named "Zeus", Fate existed: (it was) the Will of God, eternal and everywhere. But when it was named "Zeus", it was (mistakenly) supposed that it came into being, whereas in fact it (always) existed and had simply not been named previously. This is why (Orpheus) says, "Zeus first was born". For at first, Will was "Fate"; later it was consecrated as "Zeus". But people who do not understand the text imagine that Zeus is the "first-born" god...

Aristophanes helpfully satirises this style of abstract rationalisation in a few places: in the Clouds, he casts Sokrates as one of the new-fangled philosophers who engage in this kind of reductionism. Lines 319ff. (using Ian Johnston's translation because I'm lazy):

SOCRATES O Sovereign Lord, O Boundless Air,
who keeps the earth suspended here in space,
O Bright Sky, O Sacred Goddesses --
the Thunder-bearing Clouds -- arise,
you holy ladies, issue forth on high,
before the man who holds you in his mind.

And later, at 825ff.:

STREPSIADES You see how useful learning is?
Pheidippides, there's no such thing as Zeus.

PHEIDIPPIDES Then what is there?

STREPSIADES Vortex now is king --
he's pushed out Zeus.

Similarly, a couple of decades later in the Frogs we find a caricature of Euripides (another person identified with the rationalist approach) praying to a satirical set of rationalist "gods" as follows:

DIONYSUS [to Euripides] It's your turn --
take some incense. Make an offering.

EURIPIDES All right --
but I pray to different gods.

DIONYSUS Personal ones?
Your very own? Freshly minted?

EURIPIDES That's
right.

DIONYSUS Then pray away to those private gods of yours.

EURIPIDES O air, my food, O pivot of my tongue,
O native wit, O nose that smells so fine,
whatever words I seize upon, let me
refute them -- let the victory be mine.

Quoting from comic plays doesn't prove anything, exactly, but it helps to illustrate the kind of thing that carried verisimilitude at the time. Whether the ideas in these satirical quotations are associated with any specific thinker or not (but the similarities to the Derveni papyrus are certainly striking!), they're indicative of what people imagined this kind of "philosophy" did.

In the specific case of Sokrates, one of the accusations levelled at him was that of introducing new gods -- the specific charge was asebeia (impiety), of which a few naturalist philosophers had previously been convicted (including the infamous atheist Diagoras of Melos, who some think is the author of the Derveni papyrus). Both he and the Derveni author would be able to claim, to sympathetic ears, that they were not introducing new gods but reidentifying the nature of the gods. But that wouldn't, and didn't, hold water with the Athenians of the time. Outside very, very esoteric circles, there's no way that the theology in the Derveni fragments I quoted above could pass for "typical Greek gods".

Aristotle, whom you also mention, is a separate case. The interpretatio philosophica I've talked about above really only applies to the latter part of the 400s BCE. Aristotle lived nearly a century later, and the situation was very different then. I can't comment further on Aristotle, I'm afraid, as his thought requires a great degree of expertise that I don't have, except to say that absolutely nothing in my post applies to him.

[Previous threads] (http://www.reddit.com/r/AskHistorians/comments/1v8ui6/what_was_the_fallout_if_any_of_socrates_being/) have discussed Socrates' trial and death. The top comment doesn't directly answer your question about Greek philosophers belief in gods, but it does show how public opinion turned against Socrates. It suggests that the trial was more about politics than religion.

Its difficult to pin down exactly what Socrates believed on any given subject—nearly everything we know about him comes from Plato, who uses Socrates as a mouthpiece for his own ideas, especially in works like the Republic (the Wikipedia page on the “Socratic Problem” gives more info).

Plato’s metaphysics, however, are distinctly un-pagan. So much so, in fact, that Plato’s metaphysics was a huge influence on the development of Christian theology. St. Augustine of Hippo was particularly influenced by him, but, as pointed out by W.T. Jones, the entire philosophical frame work of Platonism fit early Christianity perfectly.

Aristotle, Plato’s greatest student, posed more problems for Catholic theology, particularly given the Platonic bias given the Church by St. Augustine. The thought of St. Thomas Aquinas is largely an attempt to reconcile Catholic theology and Aristotelian metaphysics. To a large extent he succeeded. One of the main issues St. Thomas had to address was that Aristotle held that there was a “necessary being”, i.e. God, but that this being’s very nature precluded any intimate connection with the created world ( similar to a deistic, clockmaker God). I won’t go into the details, but to a large extent St. Thomas succeeded in saving Aristotle for Catholicism.

So, in answer, the Greek philosophers were free thinkers who refused to accept any orthodoxy based solely on authority, including the Gods and Goddesses of Mt. Olympus. As a side note, other Greek philosophers rebelled even more radically. Google Epicurus, Empedocles and Pythagoreanism for more information.

Sources: The Classical Mind and The Medieval Mind by W.T. Jones.