Just as Act III ends, right after Antony's funeral speech, Rome erupts into a chaotic, mob-led brawl. The people have been so inflamed by Antony's speech and the murder of Caesar, that chaos reigns and the country is on the brink of a civil war.
Act IV, by contrast, opens in what appears to be a calm and controlled meeting of the three men who will become Rome's second triumvirate of rulers: Marc Antony, Octavius Caesar, and Lepidus. They are preparing to fight the Conspirators who are outside Rome, preparing to challenge the triumvirate's right to take power. The balance of the play concerns this military struggle, a struggle won by the Triumvirate, and one that ends in the suicides of both Brutus and Cassius.
In the opening scene of Act IV, Shakespeare shows a new side of Antony. Now that he is in a position to gain power, he appears much more analytical and cold, even somewhat backstabbing. He sends Lepidus on an errand (almost as he would send a servant), and while Lepidus is gone, confides to O. Caesar that Lepidus is not really fit to rule. Shakespeare uses this scene to demonstrate what the acquisition and maintenance of power can demand of a person -- acting from a non-emotional, and potentially duplicitous perspective.
Act IV, scene ii mirrors scene i, since it shows the jockeying for power going on within the Conspirators' camp. Cassius is attempting to discredit Brutus and gain more power for himself, while Brutus is clinging desperately to the moral, just ideals that he has followed up to this point. Shakespeare seems to be showing that the man who is too goody-goody (Brutus) and the man who is too hot-headed (Cassius) are both destined to fail in the struggle to gain power.
Both the play and history show that, ultimately, it is Octavius Caesar who wins the day, becoming Rome's first Emperor. But you'll need to read Shakespeare's Antony and Cleopatra to find out how that happens.