He was offered a crown three times at the Lupercalia by Antony, and turned it down. Some people think it was a staged procedure, others that he really rejected it.

Suetonius reports

But from that time on he could not rid himself of the odium of having aspired to the title of monarch, although he replied to the commons, when they hailed him as king, "I am Caesar and no king,"

That's a joke in Latin. It doesn't work in English. Rex was a normal name for certain aristocratic families.

So take your pick. He either didn't want to be king, or he staged an elaborate show to get people to demand him to be king. My personal opinion is the first - he didn't want it.