Historically, Octavius Caesar was the great- nephew of Julius Caesar; he became the first Roman Emperor in 27 B.C.E. In Shakespeare's Julius Caesar, Octavius is a relatively minor character who appears in the last two Acts of the play.
The first mention of Octavius comes at the end of Act 3, scene 1, when Octavius's servant arrives to inform Antony that Caesar had previously sent for Octavius, and that Octavius is within 20 miles of Rome and will arrive there soon.
When audiences first meet Octavius in Act 4, scene 1, he is meeting with Antony and Lepidus to discuss their strategy against the conspirators. In his first line of dialogue, Octavius tells Lepidus that Lepidus's brother's name is being added to a list of names of men who must be killed (presumably because they pose a threat to this new triumvirate). This seemingly heartless declaration, made by Octavius in such a matter-of-fact way, gives many audiences an unfavorable first impression of Octavius. However, Octavius's commitment to avenge his uncle's death is clear from his first line in the play. Octavius soon softens, though, in his defense of Lepidus when Antony refers to him as a "slight, unmeritable man,/ Meet to be sent on errands."
More significantly, audiences witness Octavius's commitment to avenging Caesar's murder during the Act 5, scene 1 battle against Brutus:
Look, I draw a sword against the conspirators. When think you that the sword goes up again? Never, till Caesar's three-and-thirty wounds Be well avenged, or till another Caesar Have added slaughter to the sword of traitors.
After Brutus's death, though, Octavius is able to objectively recognize Brutus as a noble, honorable soldier. First, Octavius vows to take all of Brutus's followers into his own service (rather than punishing--or even killing--them), and finally, with his final lines, which are also the final lines of the play:
According to his (Brutus's) virtue let us use him, With all respect and rites of burial. Within my tent his bones tonight shall lie, Most like a soldier, ordered honorably. So call the field to rest, and let's away To part the glories of this happy day.
Ultimately, though Octavius plays a relatively small part in the action of the play (and certainly if one counts the number of lines he actually speaks), Octavius's character serves as a reminder that while Caesar was unjustly murdered by Brutus, Brutus was an honorable, noble man who did what he believed was right for Rome. From Octavius's final lines, audiences can assume that Brutus will be given a proper burial, and that Octavius, Antony, and Lepidus will do their best to reestablish peace in Rome.