You ask a very complicated question predicated upon modern assumptions of the nature of homosexuality. If you want a real, but difficult-to-read source, you get Louis Crompton's "Homosexuality and Civilization".
I'll divide the conversation between VERY superficial (and I welcome better specialists to expand or correct) discussions of the concept of what you'd call homosexuality in first Greece, then Rome.
The Golden Age Greek culture takes its roots from the patriarchal Mycenaean culture it idolized in its epics. The cultural roots of Attic sexuality envisioned women as essentially base and non-virtuous: sex with women was necessary for procreation, but was always tainted with the sense of depreciating manly virtue. There are long arguments about the extent of pederastic relationships, most notably in the writings of Plato, who contended that non-sexual pederastic relationships enhanced virtue and manliness, while carnal relations were "brutish". So how do you conceive of that in relation to modern ideas of homosexuality? It's a totally different mindset. Not to mention Xenophon's almost blase criticism of the routine stationing of Spartan troops with their homosexual lovers to increase fidelity, where he believes duty to country should prevail. We get the sense, as historians, that homosexual relations were a binding thread based on the inherent supremacy of men over women in Golden-Age Greek culture.
For the Romans, the story is more complex. Monarchic/Early Republican Rome seems almost "Puritanical" in its regulation of sexual roles, adopting none of the Greek attitudes. As we get into Late Republic/Empire, what we see is a distinction based almost entirely upon the role played within sexuality. Yes, I know I'm simplifying thousands of pages of great historiography here. Suffice to say that the language preserves the Roman attitude toward homosexuality. To be in the penetrative role essentially made you masculine, and while it was vaguely questionable, did not impugn one's reputation. To be penetrated, however, made you into something effeminate, and destroyed your masculinity. In oral sex, for example, irrumatio meant the act of fucking someone's mouth. This is a masculine act. Fellatio, however, implies sucking dick, and that's feminine and for a man, utterly demeaning. It was not accusations of homosexuality that Romans threw against their enemies, but accusations of "bottoming" or receiving penetration that really damaged reputations. Most famously, Caesar spent much of his life fighting off accusations that he earned favors from a certain King Nicomedes of Bithynia by taking it up the ass from him as a young boy.
TL;DR: What we would call homosexuality seems to have been extremely common in Ancient Greece and Rome, but cultural attitudes meant that their standards for moral judgement were completely different from ours.
I can't speak to Greece (and we should probably separate pedastry from homosexuality for the purposes of that discussion) but Roman sexuality didn't fall into the next homosexual / heterosexual divide that we expect in the 21st century.
Romans were much more concerned with those who penetrated bland those who allowed themselves to be penetrated. In other words, while it would not be unusual or unseemly for a high ranking Roman to enjoy the company of both men and women, it would be considered unmanly (or, more appropriately, not in keeping with the masculine ideal of the time) for him to perform oral sex upon another man or allow another man to penetrate him. Thus the male partners of powerful Romans tended to be boys of significantly lower social rank.
There is an excellent pop history treatment of this in "Holy Shit" which delves into the nature of Roman swearing and curse words, specifically the offense given by threatening to force someone to perform oral sex.
To my understanding, sexuality wasn't a binary between hetero/homo like many view it today.
u/heyheymse is an expert on this topic: http://www.reddit.com/r/AskHistorians/comments/136omi/wednesday_ama_i_am_heyheymse_specialist_in_roman/ http://www.reddit.com/r/IAmA/comments/u7zpn/iama_heyheymse_from_askhistorians_i_have_a_degree/