I'm wondering from several perspectives. His full name was Gaius Julius Caesar, but in modern times, few people know him as anything other than "Julius Caesar".
How would his friends, colleagues, subordinates, enemies, etc addressed and referred to him? Would those close to him have used the name "Gaius" as a given name, the way the western world generally uses first names today? Was his name so ubiquitous that people of the time would simply have said "Caesar", even to his face? And what about the name "Julius"? Does that come from gens Julia, Caesar's Patrician family, or is that also a given name? And if it is a family name, where did the name "Caesar" actually come from?
Basically, how did Roman names work?
Found this: http://www.vroma.org/~bmcmanus/roman_names.html
So close friends like Mark Anthony could address to him "Gaius" in conversations, and any senate coleagues, supporters or enemies, addressed as "Caesar".
"Julius" is a family name. He had one personal name "Gaius" and two family names "Julius Caesar", like most patricians.
Julius Caesar claimed that his family name "Caesar" meant "elephant" in some other language and that cognomen was given to (or taken by) his ancestor because he killed one. Caesar even minted coins with the picture of elephant.