The 'Roman Empire' was the long period of history of Rome stretching over some 500 years, which began in the 'post-republican' era i.e. after the end/fall of the Roman Republic and introduction of dictatorship and later Emoire.
The initial or 'motive event' which marked this transition of change, was, probably, the election or appointment by the Roman Senate of the dictator Julius Caesar as 'perpetual dictator' in 44 BC-- a step which in effect was like commiting suicide, as the senate and republican organisations lost all powers thenceforth. Caesar was assassinated by a group of diehard Republican Romans in March of the same year i.e. 44 BC, and thereafter Rome was plunged into civil war, between the Republicans and 'Caesarians' or Imperialists-- Caesar had, prior to his demise, declared his adopted son (nephew originally) Octavius/Octavian (later to become the Emperor Augustus Caesar) as his legal heir and successor and this was a rallying point for supporters of 'Caesarism'; and at the final and decisive Battle of Actium, September 31 BC, Octavius/Augustus and his supporters won. Octavius and his supporters (including Mark Antony and others) carried out several severe reprisals for the murder of Julius Caesar and this had a terrifying effect on opponents.
A much-chastened Roman Senate, now by and large only a 'rubber stamp', granted formally the title of 'Imperator' and 'Augustus Caesar' unto Octavius/Octavian in January 27 BC. This was the initiation of the Roman Empire.
Many of the early events of the time, from the death of Julius Caesar and to the 'founding' of a dynasty of Caesars, are also given in Shakespeare's play Julius Caesar.