I was unable to find an answer to this question in the search engine of this subreddit, so I apologize if this question has already been answered.
Julius Caesar's own life is well documented, but what of his parent's? How did they raise him? In regard to his families lineage, where and when was the birth of the Caesar name?
I can't speak for his parents, but his ancestry was one of the oldest in Rome. His middle name, Julius, means that he was from the House of Julia. This was one of the most ancient houses in Rome. This was actually important in the life of the great Caesar himself. This is because at funerals, people played up the noble lineage of their family. He claimed he and his family descended from Venus, whose son Anchises fled from Troy after the Trojan war.
Livy also tells a story that Julius Proculus, who informs the Romans that Romulus had ascended to heaven, and had given him a divine message. "Go, he said, and tell the Romans that by heaven's will my Rome shall be capital of the world. Let them learn to be soldiers. Let them know, and teach their children, that no power on earth can stand against Roman arms."
In all likelihood, amidst the legend, it seems that the family came to Rome during the reign of Tullus Hostilius, the third of the seven Roman kings. This would make the name very ancient and noble, even without the legends surrounding the founding of Rome.
The name Caesar, on the other hand, was a family name. It tended to hold less gravitas than the house name. You don't see the name Caesar held in very high regard until Caesar himself comes along. The earliest record we have for a Caesar was Sextus Julius Caesar, who was praetor in 208 BC. Like many patrician families, the Caesar's had dealings in nearly every great event that took place after that, but playing a more subsidiary role. They tended not to have the same auctoritas or gravitas as a Scipio may have.
In short, the name Julius was an ancient and powerful name, but Caesar is what made the name Caesar great.
According to Pliny the Elder in his Natural Histories (7.54), Caesar's father bore the same name and held the rank of Praetor, the second highest rank in the Roman Republic under Consul, died when Julius was only 16 (Suetonious, Lives of the Cesears, Life of Julius Caesar 1.1).
~~As far as I could find, no Caesar ever held the office of Consul, though the family was a long running patrician family from the Julian line.~~edit: I didn't look hard enough.
I cannot seem to find any material that actually discusses his mother, though in Plutarch's Life of Marius (para. 6) and Life of Caesar (para 5.), Caesar's is mentioned to have an Aunt Julia that marries Marius.
A small dialogue named Dialogus de oratoribus 28 attributed to Tacitus names Aurelia Cotta as the Mother of Julius Caesar. It praises her as being the ideal kind of Roman woman.