Octavius does not really develop all that much as a character in Antony and Cleopatra. The main reason for this is because he has been seized by the utterly ruthless, single-minded pursuit of power. Nothing else matters for Octavius; not love, not friendship, not anything; all he wants is to become master of Rome. This makes him a foil to Mark Antony, who prioritizes his love for Cleopatra over the acquisition of power. Antony's passion for the exotic Egyptian queen makes him a figure of ridicule and suspicion in Rome. In his bitter power struggle with Antony, Octavius presents himself as the epitome of Roman virtue fighting against a man seduced and corrupted by the luxurious trappings of Egyptian court life.
If anything, Octavius becomes more ruthless, more single-minded, and more amoral in his quest for power as the play progresses. It seems that there are no depths to which he is not prepared to stoop in order to fulfill what he regards as his destiny. We see this when Dolabella warns Cleopatra that Octavius plans to have her captured and taken to Rome in chains. Cleopatra's subsequent suicide is her way of avoiding such a humiliating public disgrace.