On March 15, 44 B.C, Julius Caesar was killed on the floor of the senate shortly before he was to be crowned emperor. Still today, many people debate on whether the killing of Julius Caesar was just or not. I'm wondering if the senate was permitted by law to crown Caesar Emperor, because this seems like it would be a notable point in an argument towards the un-just side.
Caesar was never crowned emperor. The beginning of the empire is dated to Augustus' adoption of certain legal powers (notably tribunate power) while holding no official legal magistracies, because it meant that the individual of the emperor held legal power outside of the normal legal means. Caesar held sole power very much by legal means, and is not considered to be an emperor. Caesar was murdered after (not before, as you say) Mark Antony's act to confer upon him the office of the dictatorship without limitation of term in office (i.e. for life). The dictatorship was a perfectly legal magisterial office, elected by the citizen assemblies or appointed extraordinarily by the senate (and no crown was involved whatsoever--Antony, as Caesar's colleague in the consulship, offered Caesar a diadem at the Lupercalia in 44 and was rejected). The office had fallen out of use after the Punic Wars, being briefly revived by Sulla, and Caesar's dictatorship for life was in fact his third dictatorship. All of them were perfectly legal, if not exactly normal--in fact, they were arguably more legal than Cato's proclamation of Pompey as sole consul of 52, which was made at the declaration of the senate, who technically was not supposed to be able to appoint consuls, who were popularly elected. What might not have been entirely legal was that Caesar held the consulship in 44 while he was still serving as dictator. However, he had done this before--his third and fourth consulships, in 46 and 45, had both been held while he was serving as dictator for those years--and Sulla had served as dictator immediately prior to his second consulship, which Caesar also did during his first dictatorship. And legally there was no law that said that a dictator couldn't be a serving magistrate, it had just never been done before