The first Roman emperor was Augustus, Caesar's adoptive son. Augustus was born as Gaius Octavius, though in the English world we know him as 'Octavian'. Caesar adopted Octavian as his son in his will. After Caesar was killed, Octavian spent two decades first fighting Caesar's enemies and then his own enemies. Having emerged as the sole victor after decades of civil war, Octavian was granted the titles of Augustus ("Illustrous One") and Princeps ("First") in 27 B.C. As a victorious general, he also could use the title Imperator, which is the word where the English word 'emperor' is derived from. It are these titles which were (mostly) used by all emperors following Augustus. This is why we consider Augustus to be the first emperor.

Caesar did not have these titles; Caesar did not even ask for these titles. When Octavian was granted the titles Augustus and Princips his position was a lot more secure than Caesar's position ever was. Octavian maintainted the pretence of the Republic while subverting it with his new titles, but Caesar could not be that bold. All Caesar could do was usurp existing mechanics in the Republic. Caesar served as consul (the highest regular office in the Republic) several times in a row, even though that officially wasn't allowed. Near the end of his life Caesar secured to position of dictator, an extra-ordinary position giving someone sole rule over Rome, traditionally for only 6 months. Caesar though was granted this position first for two five-year periods and then for life. At this point his enemies decided he had become too dangerous for the Republic, so they killed him.

So this is why Caesar was not an emperor: Caesar abused existing positions within the Republic (as others had done before him, perhaps not to the same extent). Octavian/Augustus created a new position with new titles, effectively ending the Republic and starting an empire.