The most obvious way Ovid reveals his characters is through their actions. Thus, for example, the fact that Icarus ignores his father and flies close to the sun shows that he is reckless and impetuous. Similarly, by giving Jupiter and Mercury everything they have, even though they have basically nothing, Baucis and Philemon reveal their profound generosity.
Ovid doesn't use clothing all that often as a tool of characterization, but there are some special cases. For example, Atalanta's simple hairstyle and bow-and-arrow accessories express the fact that she is an athletic tomboy.
Throughout his poem, Ovid uses a lot of direct characterization to reveal his characters. This can be either statements about their thoughts, their actions, or their whole way of life. For example, in Book I, Ovid introduces Deucalion by saying that he is the most just man on earth. Then he tells us that Deucalion's wife Pyrrha is the most pious woman. There are lots of moments like this throughout the poem.
Unfortunately, even a nice guy like Ovid has a few cultural prejudices. If you happen to be from Thrace, Ovid will automatically assume that you are a sex-crazed barbarian who won't take no for an answer. Sorry.
One important way of classifying characters in Ovid is by divine status. His poem contains tales of straight-up gods (Jupiter, Juno, Minerva, and so on), lesser gods (river gods and such), demigods (like Hercules), and mortals (like, say, Deucalion and Pyrrha, or Daedalus and Icarus). It would be hard to generalize and predict someone's personality based on what kind of being they are, but the power of Ovid's gods must always be taken into consideration – which can make even their slightest whims have very serious consequences.