Why did Shakespeare choose to write about Julius Caesar?
I believe that there is a further answer to this question which lies in the politics and recent history of the times in England. Shortly before writing Julius Caesar, Shakespeare had completed his cycle of plays dealing with the history of Elizabeth I' s ancestors who had wrested the throne from Richard II which resulted in the War of the Roses, a series of Civil wars between rivals for the throne. Obviously since Shakespeare was dependent on patronage from the crown and her supporters, Shakespeare's portrayal of the Queen's ancestors was positive showing them to hold the moral high ground. By setting a story of royal revolt and murder in in a distant time and place, Shakespeare could comment on the evil of revolt against a blameless ruler and the resulting civil wars without angering Elizabeth. Thus I believe, Shakespeare used Julius Caesar to make a non threatening comment about the political questions of his day
For many living writers, we have extensive records of their thoughts about their works. We have journals, interviews, even websites. We don't have those for Shakespeare. For the most part, we must deduce his intentions and reasoning from the works themselves.
If we do that, and we look at Julius Caesar as a subject matter in context, we get several reasons he might write about him. First and most simply, it's a juicy subject matter for plays. Second, and almost as simply, there was a long tradition of writing tragedies about the upper classes—think of Hamlet, who is a prince, and Lear, who is a king. There are, however, two more likely reasons. One is that Shakespeare invented very few of his plots, working instead from fairly known source works or historical stories. In this case, his source was likely to have been Plutarch's Lives. The final and most pressing reason is that there are elements of Julius Caesar's life that echo contemporary politics, and using a distant figure with the same concerns was a way to comment on current politics without cutting too close to home.