Augustus was the first Roman emperor. The major difference between Julius Caesar and Augustus is that Caesar held his power within the framework of the Republic. Every one of the positions Caesar held - even the one of dictator with no time limit - was held by someone else before him.

Unlike Sulla (the previous dictator with no time limit) though Caesar showed no signs of stepping down after his work was done. Some senators thought Caesar was acquiring too much power and murdered him. This sparked a series of civil wars which in the end were won by Octavian, Caesars heir. During this time Octavian held several positions, also within the framework of the Roman Republic.

In 27 BC Octavian laid down all his official functions in the Roman Republic, giving power back to the senate and people of Rome. For "restoring the Republic" after decades of civil war, the Senate granted Octavian the titles Augustus ("Illustrous One") and Princeps ("First"). The Senate also decided to give Augustus the governorship of Spain, Gallia, and Syria (this was likely orchestrated). These provinces just happened to be the ones where many legions were stationed, so with this single governorship Octavian had control over the majority of Rome's legions. Octavian also got tribunal powers, without being a tribune (or even being eligible for the tribunate).

It is the unique concentration of various formal and informal powers which is the defining feature of Roman emperors. For a couple of centuries at least, the charade of the emperor simply being someone helping the Senate and "Republic" was kept up. Augustus was the one to implement this system, and he was the first one to hold these powers and titles. Augustus' titles - Imperator, Augustus, Princeps - and family name - Caesar - would become the titles and name used by nearly every emperor.

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I don't believe there's any real debate about what I said above. Some people might argue that Caesar already made some moves towards the changes instituted by Octavian, and there probably is a case to be made there, but overall it is accepted that the Republic ends with the civil wars after Caesar's death and thus that Caesar was not an emperor.

If you have a look at the Companion to the Roman Army (edited by Paul Erdkamp, published by Blackwell, 2007) it is divided into sections. Part II "Mid and Late Republic" covers Caesar and has events up to and including the civil wars. Part III is even more explicit; it is called "The Empire (Actium to Adrianople)". "Actium" refers to the battle of Actium, which ended the civil wars. The very first article in that section is on the Augustan reforms of the army. Same goes for the book A Short History of Roman Law (Olga Tellegen-Couperus, Routledge, 1993). There part II is literally "The Late Republic (367-27BC)" and part III "The Principate (27BC-284)".

Another book, The Cambridge Companion to the Roman Republic (edited by Harriet Flower), includes a chapter on the 'crisis of the Republic' and Caesars actions. Same goes for the chapter "Military Organization and social change in the later Roman Republic" by John Patterson in the book War and Society in the Roman World (edited by John Rich and Graham Shipley, published by Routledge, 1993). That too includes Caesar. The book The Roman Empire; Economy, Society and Culture by Peter Garnsey and Richard Saller (Univ. of California Press, 1987) starts with Augustus.

I think you get my point by now. It is completely uncontroversial in the academic world that Caesar lived and operated under the Roman Republic and that the Roman Empire started while Octavian/Augustus was in power.