Why did William Shakespeare write Julius Caesar?

To address this question it's necessary to look at the political background against which Shakespeare wrote Julius Caesar. Queen Elizabeth I had been on the throne for forty-one years. Under her rule, England had reached a zenith of power, wealth, and influence. The Queen was widely hailed as Gloriana, a wise, tough-minded monarch, both loved and respected by millions.
At the same time, however, it was notable that Elizabeth had not openly named her successor. Many people at the time, including Shakespeare, thought that the Queen's silence on the succession issue was causing much instability within the realm. There was a genuine fear among many that the handover of power upon Elizabeth's death would be far from smooth; indeed, the kingdom might even experience the bloodshed and chaos of civil war. The prospect was simply too terrible to contemplate.
So Shakespeare, in writing Julius Caesar, was drawing the attention of his fellow Englishmen to the potentially dangerous consequences of the death of a great leader. Even though Brutus and some of the other conspirators genuinely believed that they were overthrowing tyranny, their actions led to the establishment of an even greater one under Octavian, who later became the Emperor Augustus. In Julius Caesar, Shakespeare cautions his fellow countrymen not to rock the boat, not to allow their political ambitions to get the better of them and potentially usher in an extended period of civil strife which could well lead to dictatorship.