In the immediate aftermath of the assassination, a deal was made between the leaders of the assassination and Marcus Antonius, Caesar's right hand man (and effectively the new leader of the Caesarean faction) to not seek another civil war or more murders. The assassins/liberators would honour Caesar's legislation and Antonius wouldn't try to seek revenge. So into this very tense situation rode Gaius Octavius, the adopted son of Caesar. He had been studying in Greece when he found out that he was Caesar's heir and against the advice of everyone around him, he rode back into a snake pit.

Cicero, who detested Antonius, jumped on Octavius hoping to manipulate the boy to support the assassins/liberator faction against Antonius and also split the Caesarean faction. The goal being to use his name as Caesar's heir until Antonius was destroyed and then toss him on the scrap heap. And Octavius, probably the smartest teenager in history, sided with the assassin/liberator faction. He joined an army that marched north to confront Antonius who Cicero managed to convince the senate to declare an outlaw. In the battle, Antonius is defeated and retreated with his army into Gaul where another ex-Caesarean governor sat with an army (Marcus Aemilius Lepidus). The two consuls in the senatorial army had died in the battle or shortly afterwards. That left Octavius as the sole leader who was using (as was Roman custom) the name of his adopted father as Gaius Iulius Caesar Octavianus and so he snapped up the entire army.

He demanded the senate name him as a replacement consul and when they did not he marched his army into Rome and had himself elected (the populace generally liked Caesar and no one in the senatorial party was going to say boo to a man who had armed veteran soldiers on the streets of Rome). So now officially imbued with the powers of a consul, he then made an alliance with Antonius (who had allied with Lepidus) and they formed the Second Triumvirate. And promptly proceeded to murder their enemies, revenge for Caesar's assassination was back on the menu (Cicero got the chop for all the vitriol he directed at Antonius).

The surviving assassins/liberators and their senatorial supporters assembled in Greece and built up an army. Antonius and Octavianus landed an army in Greece and defeated the Brutus and Cassius at Philippi. They then partitioned the Roman empire between them. Antonius would have the rich Greece east, Lepidus would get Africa (which supplied more than half of Rome's grain giving him essentially a "kill" switch on the city), and Octavianus got Italy, Gaul, and Spain. The two older men thought they were handing a poisoned chalice to the teen since they now had tens of thousands of veterans who needed to be demobilized and expected to be rewarded for their years of service. So Octavianus how to figure out how to find land for thousands of soldiers who would not be best pleased if all they got was a handshake and a speech. The young leader had either to expropriate land from citizens which would be enormously unpopular and could trigger an uprising or disappoint the soldiers that could also trigger an uprising. He opted for the former and had to put down revolt of disaffected and newly landless Romans (led by Antonius' brother and wife no less).

So things continued for a few more years with this tripartite division of the empire until 36 BCE when Octavianus and Lepidus joined forces to crush the piratical Sextus Pompeius (son of Pompeius Magnus who'd been a thorn in the Caesarean faction's side ever since his father's death) in Sicily. After crushing Pompeius' fleet, Lepidus underestimated Octavianus and tried to get the latter's legions to defect to him (which would have effectively removed him from power). Instead in a baller move, Octavianus marched into Lepidus' camp and won his legions over which effectively left the other triumvir powerless. Instead he was exiled and Octavianus now took over Africa which meant Rome's food situation became much more secure.

What followed was a cold war between Antonius in the east and Octavianus in the west. Octavianus had propaganda spread about how Antoninus was enslaved by Cleopatra. When Antonius captured Armenia in 34 BCE, he made his son ruler there. Octavianus the following year as consul viciously attacked Antonius in the senate. Things got more heated and senators started to take sides, some moving east to be with Antonius. Octavianus broke into the Temple of Vesta to grab Antonius' will and revealed that he had given some Roman controlled territories in the east to his sons to rule and that he intended to be buried in Alexandria. Well clearly he was not behaving like a proper Roman and obviously under the spell of that Cleopatra and so shortly afterwards Octavianus had the senate declare war on her (see, he wasn't starting another Roman civil war, he was fighting Cleopatra!) and that would lead to Actium and Octavianus becoming sole leader.

So Rome had leadership during that time, early after the assassination the senate was able to assert itself (under the leadership of Cicero) briefly before the orator's bold gamble blew up in his face and Octavianus, Antonius, and Lepidus split up the Roman world between them. The power struggles continued with Lepidus being shoved out of power and eventually another civil war broke out between the remaining triumvirs over supreme power.