How does the political sitution in Rome following Caesar's murder compare with what it was under Caesar's rule?
Quite hilariously, really. The conspirators spend the first two acts of the play orchestrating the assassination with the most care and attention of (I think, at least) any single act in Shakespeare. Cassius has a properly and clearly orchestrated plan to get Brutus on board (letters at his window, carefully placed persuasive words, appeals to his honour, turning up at his orchard in the middle of the night). They endlessly argue about how to do it, who should be involved, who should be killed, and so on.
And yet they make absolutely no plans as to what happens next. We don't see it on stage, but we are told about the chaos that breaks out after the murder:
Men, wives and children stare, cry out and run
As it were doomsday.
And then, after Antony has turned things around at the funeral, it gets even worse:
Come, away, away!
We'll burn his body in the holy place,
And with the brands fire the traitors' houses.
Take up the body.
Go fetch fire.
Pluck down benches.
Pluck down forms, windows, any thing.
I heard him say, Brutus and Cassius
Are rid like madmen through the gates of Rome.
Before, Rome was ruled by one man, Julius Caesar. He was old, losing his hearing a little, and not in the best of health. But there's no evidence (unless you count the murder of Flavius and Marullus 'for pulling scarves off Caesar's images' early on in the play) that he's tyrannical, although plenty of characters seem to think so.
After his murder, chaos reigns for a while. But Rome falls into the hands of the triumvirate: Antony, Octavius Caesar, and Lepidus. Yet, as you see from the very carefully placed Act 4, Scene 1, the triumvirate immediately start to disagree. It's clearly not going to be an easy ride. It's a totally different leadership system.
But the play doesn't really focus too much on Rome (only really on the battle of Philippi) after Caesar's murder. If you want to find out what happened next... read Antony and Cleopatra!