Did Caesar deserve to die?

Though an argument could probably be made for both sides, I would argue that the character Caesar did not deserve to die in Julius Caesar. Here, it is important to differentiate the character Caesar from the historical figure. Though many acknowledge the threat the historical figure Julius Caesar was to the republic of Rome, Shakespeare writes the character of Caesar more ambiguously. 

For one, he is plagued with all sorts of physical issues. He and Calpurnia haven't been able to have children, which he blames on her when he tells Antony to whip Calpurnia when he runs by to cure her barren womb. Of course, it's equally possible that Caesar himself is sterile. Additionally, the audience hears from Cassius that Caesar is a weak swimmer, having almost drowned in a swimming competition, and that he was very sick as a young man. Add that to the deaf left ear and epileptic fainting in Act 1, scene 2, and Caesar doesn't look like such a huge threat anymore. 

Furthermore, Shakespeare doesn't make it clear that Caesar really wants to be king of Rome in the first place. Yes, there is that whole theatrical production in Act 1, scene 2, when the crowd offers Caesar a symbolic crown and he refuses it three times. Still, that seems more about Caesar loving the attention and love of his people than actually trying to take over Rome. 

While there are moments of the play and dialogue that suggest that Caesar could be a threat to Rome (particularly his Northern Star speech right before he is assassinated), he still did not deserve to die. After all, the conspirators who planned his death did so for the good of Rome and the will of the people. But these same conspirators mock "the people." When relating the people's reaction to Caesar's weird behavior at the crown offering, Casca says the people

"cried 'Alas, good
soul!' and forgave him with all their hearts; but
there's no heed to be taken of them; if Caesar had
stabbed their mothers, they would have done no less" (Act 1, scene ii).

Aside from ignoring and mocking the preferences of the people, the conspirators' claims of fighting for the republic ring false due to Cassius's scheming. The audience sees how aggressively he manipulates Brutus, even sending fake letters to his house, pretending that the people of Rome sent them. Caesar himself identifies the "lean, hungry look" of Cassius, recognizing him as a discontented schemer. In fact, much of the information the audience gets about Caesar comes from the conspirators themselves, who are all already against him. When viewed this way, the plot seems to be less about the freedom of Rome and more about Cassius' ambition. There just does not seem to be enough objective evidence that Caesar is trying to take over the power of Rome for his death to seem just.