Because this is a play, the most obvious way that we learn about the characters is through their actions. Thus, when Orestes makes an offering of two locks of hair at the tomb of Agamemnon, we can tell that he is loyal to the memory of his father. When Electra and the Chorus of women come and make their offerings, we can tell that they are obedient to Clytemnestra who sent them. When Chorus goes on to pray for the return of Orestes, we also see their disobedient side (towards Clytemnestra and Aegisthus), and their deeper loyalty (towards Agamemnon's memory).
It's important to bear in mind, of course, that actions can also be deceiving. When Clytemnestra receives the disguised Orestes and Pylades with great hospitality and pretends to be sad at Orestes's death, this conceals the fact that she might actually prefer him dead. Thus, in judging a character based on his or her actions, it's often safest not to judge until the end of the play, when you can look at everything they've done, not just things here and there.
We also get a lot of direct characterization in Aeschylus's play, not from the narrator (there is none) but from the characters, who either talk about themselves or about other characters. For example, from what the Chorus and Electra say in the opening scene, we can tell that Aegisthus is a real jerk. From what Orestes says about himself, we can tell that he is dedicated to the cause of revenge, even though it ultimately leaves him tormented by guilt. Of course, because any character's opinion about himself or herself – or about another character – is inherently biased, any direct characterization in the play must be weighed against what we actually see of the various characters' actions. That's the only way we can understand what Aeschylus really wants us to think about the characters. In looking at characters' actions, the same warnings from above apply.