From reading your question it looks like you're confused on the chronology - Julius Caesar ruled before Augustus (who is usually counted as the first Roman Emperor). So that's that. But now for you possible next question: why is Augustus counted as the first emperor and not Julius Caesar?

Edit: I'm guessing you mistook Julius Caesar's dates for AD instead of BC

For that I'm going to steal from an earlier answer of mine (where I actually just direct you to other answers):

The two answers in this linked thread from u/LegalAction (via a linked older thread) and u/winter0215 might also be of interest. The old Republican institutions stuck around: the Senate and tribunes, for example. So it's a little bit of an arbitrary decision to say the imperial period begins with Augustus (who as the linked answer notes actually described himself as saving the republic - whatever his actual goals and results, that was the public-facing message). I think most people are pretty content with applying the definition to Augustus' reign, but if it seems a little fuzzy to you that's because it is! Nobody ever formally dismantled the republic to give us a nice clean definition. But Augustus ruled a long time, consolidated a lot of power, and set the precedent. That lengthy reign is important too, a whole generation grew up not knowing any other ruler (he was no Queen Elizabeth in that regard but still pretty good for the first century AD!).

https://www.reddit.com/r/AskHistorians/comments/6yu2lw/how_long_after_the_rise_of_augustus_did_it_take/

This [thread] (https://www.reddit.com/r/AskHistorians/comments/786v5g/romans_regarded_julius_caesar_as_the_first/) by /u/LegalAction goes into the consensus (or lack thereof) on the question of whether Julius Caesar should be considered the first Roman Emperor. That being said, the content of your post seems to contain a misunderstanding of Roman chronology, both Claudius and Nero were emperors after Caesar’s reign and death, in fact Claudine’s was born 34 years after Caesar was assassinated.

Broadly speaking, Gaius Julius Caesar was never recognized as an emperor because such a position did not exist during his lifetime.

The Roman Republic to which Ceaser was born into was managed by two elected consuls (the highest political office) who stood for a one year term. To Rome, the greatest political crime would be to be a King. Men such as Brutus proudly traced their history back to those who assassinated Rome's last king. So a system where two elected magistrates balanced each other's power was the optimum solution.

Unfortunately, the final decades of the Roman Republic were a mess. In Caesar's early years, a civil war between Marius and Sulla erupted across the Republic, ending with the appointment of Sulla to the position of dictator. This was a legal position designed to be used in periods of crisis, which granted the individual full powers of the state, above all other magistrates (including consuls). Sulla stepped down after 3 years (in 79BC) and the consuls became the leaders of the Roman Republic once again.

To summarise Ceaser's well documented subsequent years, he was was elected consul in 59BC, led the armies of Roman in a conquest of Gaul for 10 years and returned with an army in defiance of the Republic's laws. This kick started his civil war against Pompey (one of Ceaser's former allies) at the end of which, Ceaser appointed himself as dictator.

What made Ceaser's dictatorship different to the dictatorships of Sulla and his predecessors, was the term. Ceaser was the dictator perpeto, i e. his dictatorship renewed every year. The Senate remained in existence, and although it existed in conjunction with his dictatorship, became increasingly subservient to and controlled by Ceaser.

It was not until Ceaser was assassinated and the subsequent civil war resulted in the victory of Octavian (Ceaser's adopted son) that we see the formal end of the Roman Republic.

Octavian renamed himself Augustus and the Senate made him Princeps Civitatis (the first citizen). This granted him official supremacy over the Roman Senate and made him "first amongst equals". Although the old political machinery of the Republic remained in place, the new Augustus had supremacy over it, spoke first in the Senate, and held true power over the Republic to a degree exceeding Ceaser's.

For those reasons (specifically terminology, and political power) Ceaser was not an Emperor but a perpetual dictator.