Shakespeare was mainly concerned with making Octavius appear to be different from the three men with whom he interacts. Shakespeare has already differentiated between the three other characters. Since Antony, Brutus and Cassius are all older by a generation, Shakespeare makes Octavius seem young, impetuous, pugnacious, and very anxious to assert his new status as the heir of Julius Caesar. He is not going to be treated as a junior. (Cassius calls him "a peevish schoolboy.") Octavius is presumably, like most young men, not so sure of himself as he would like to appear. He is excited at the prospect of being involved in a great battle with thousands of men being killed all around him. Not only that, but he will be one of the generals--and then he can look forward to being one of the rulers of the Roman empire. He does not, of course, think there is any possibility of being killed himself. Young men expect to live forever.) The three older men are in no great hurry to begin fighting. They have seen plenty of war already in their lifetimes and know, like General Sherman, that war is hell. Octavius seems like a young man who is anxious to prove himself. He is inexperienced in everything, and he will get into trouble on the battlefield because he is bound to make serious tactical mistakes and also because his men probably have no respect for him and no confidence in his leadership. He is really only Mark Antony's protege, but he doesn't want to be perceived in that role. He has to keep asking Antony what to do, both at Rome and at Philippi, but then he doesn't want to be a yes-man to Antony. Octavius deliberates chooses to "cross" Antony (i.e. to move his forces to the other side of the battle lines) just before the battle begins. This is juvenile behavior. It is no time to be making such radical adjustments in the battle order. He should have listened to Antony, who was a real warrior and infinitely more experienced in military matters; but he was being deliberately perverse in order to assert himself. The main character trait of Octavius is simply that he is young. He only holds such a prominent military and political position because Julius Caesar named him as his heir.
Although Octavius does not actually enter the play until Act IV, by the end he appears as one of the foremost characters. He has triumphed over Brutus and Cassius and along with Antony he now stands at the apex of Roman political and military power.
Octavius grows in stature throughout the last two acts of the play. Young, laconic and practical-minded, he initially appears somewhat in the shadow of the older and more experienced Antony but is quick to assert himself. He refuses to let Antony dictate to him on the battlefield, sending out an ominous hint as he does so:
Antony: Why do you cross me in this exigent?
Octavius: I do not cross you; but I will do so (V.i.19-20)
Octaivus, then,threatens to 'cross' Antony, to supersede him in future - and indeed,the historical Octavius did go on to overthrow Antony and amass supreme power for himself as the first emperor of Rome.
Overall, Octavius appears in this play as a man of few words and decisive action. Altogether he comes to appear as an imposing and impressive character, who also pays due tribute to his chief opponent, Brutus, at the end of the play.