We can't really make any assumptions about what people thought about the Greeks and Trojans at the time of Homer - we have little written anything from the time. Homer himself (or the Homeric poets) makes little attempt to make the Greeks seem more glorious and at times make them seem violent and cowardly in the face of the noble Trojans - at other points, the opposite is true. This relative lack of bias indicates that people in Greece listening to these poems as they were composedprobably felt sympathy and disgust for characters on both sides. A section of the 'The Iliad', 'The Catalogue of Ships', is essentially a list of which parts of greece were led by who and how many ships they came in as a contingent - we might fancy that Greeks hearing the name of there own town felt pride and thus identified with the Greeks more, since local people were invloved in their fighting. However, this is nothing more than a fancy.

From later periods, retellings/writings which indicate what attitudes towards certain characters might be.There are some portryals where you are supposed to root for the Trojans. For example Euripides, an Athenian writing in the 5th century BC, follows the women of Troy after the sack of the town in his play 'The Trojan Women', which portrays the greeks in a pretty brutal light. We also have from around this period many negative portrayals of Odysseus as deceitful and selfish. Hector, on the other hand, is almost if not always portrayed in a positive light.

Ultimately, just as it is now, whether or not you supported the Greeks or the Trojans was up to you. There are many portrayals of the Greeks in a negative light, and the Trojans tend to be viewed with respect, but for your average Joe, no one can say whether the Greeks would have been supported just because they were Greeks. There was probably a great variance of opinions at different time periods.