As in all plays, the characters in Angels are defined by their actions. Roy's manipulative tactics, underhanded dealings, and blatant hypocrisy clearly show exactly what kind of person he is. Louis reveals the weakness of his character when he abandon his sick boyfriend, Prior, although we also see him wrestle with the guilt caused by that decision. We see Joe's struggle with his homosexuality very clearly over the course of the play, and his ultimate acceptance when he gets together with Louis at the end.
Many of the characters in the play are partly defined by their sexual preferences. Roy and Joe are both deeply in the closet. Roy has liaisons with men all the time, but he tries to deny this even to his doctor, who knows better. The fact that he acts on his sexual desires but lies about it seems to highlight his hypocrisy.
On the other hand, it seems that Joe has never acted on his homosexual feelings, at least not until the end of Millennium Approaches. Though he's always recognized his sexual preferences, he was taught that they were evil and that the moral thing to do was to try to repress them. Joe has always tried to do what he thinks is right.
Many of the other characters are defined by their sexuality as well. For example, Prior, Louis, and Belize are all openly gay men – not an easy lifestyle, especially during the time the play is set. Then, of course, there's poor Harper, who's desperately in love with Joe and is nearly driven crazy by the fact that he will never desire her.
Kushner's stage directions give us some helpful tips for understanding the characters. For example, we're told that Roy is an "unofficial power broker," that Prior "lives very modestly but with great style off a small trust fund," and that Belize's real name is "Norman Ariaga; Belize is a drag name that stuck."
For the most part, Kushner lets us figure out the characters' personalities from the play itself, but we get a few hints. For instance, we're told that Prior I is "blunt" and "gloomy," and that Prior II is "sophisticated."