Since Electra is a play, you can anticipate that most of what we know about these characters comes from the things they say. Electra is always talking about duty; so she's dutiful. Orestes is always talking about action; he's a man of action. Chrysothemis dishes about pragmatism; she's a logical gal. You get the picture.
In Electra, we can get a sense of the character's ideological stance based on how they are treated by members of the family. Electra is treated like an outsider by her family, and this loner-status clues us in to her ideological isolation. Orestes is a foreigner to Mycenae; he grew up away from his blood-soaked family and is back as the redeemer or savior. Chrysothemis is treated by Clytemnestra as a dutiful daughter; she is pragmatic in her attempt to secure a good position with the royal couple.
The action or inaction of these characters informs us about their larger function in the play. While Electra is the protagonist of this play, she doesn't seem to act very much. She mostly stands around and talks, wails, or, most often, laments. It's fair to say, then, that Electra is not the one driving the action of this play. She is a great schemer, but does not herself carry out the plans. Orestes, on the other hand, shows up in town ready to act. He goes to the palace. He murders Clytemnestra and her husband. He's the one making things happen in this play – he is the man of action.
One more thing to note: Sophocles's Electra is unusual in that the relative action of a particular character does not necessarily reveal how important that person is to the overall plot. Another place where we encounter a protagonist who is hesitant, or downright unwilling, to act is in Hamlet, by William Shakespeare.
In Greek tragedy, all the parts were played by only three actors. As you can see, there are more than three parts in Electra, which meant the same guy was playing more than one part. In Electra, the man playing Orestes would also play Clytemnestra. This is interesting, since these two actors interact with Electra in such opposite ways – one as her ally and the other as her enemy.