How was Caesar able to rise up in power from an ordinary citizen to such a powerful leader of Rome?
First, we need to be clear: Julius Caesar was not an ordinary citizen. Gaius Julius Caesar was a member of the Julian family, which traced their heritage to the goddess Venus. According to mythology, Venus was the mother of the Trojan hero Aeneas, whose son was Iulus. The Julian family traced their heritage to Iulus.
Not only could Julius Caesar trace his family line back to Venus, but at a young age he was betrothed to the daughter of Cinna, who had been a Roman consul several times (ordinarily a Roman could only serve as consul once).
Before Julius Caesar had reached his mid-twenties, the Roman dictator Sulla recognized that Julius Caesar would be a force to be reckoned with (see Suetonius' Life of Julius Caesar). Caesar even had to leave Rome for a time to avoid the wrath of Sulla.
After Sulla's power began to subside, Caesar returned to Rome and made a name for himself as a lawyer, soldier, and politician. Ultimately, by the time he was in his late 30's, Caesar had managed to link himself with Rome's two most powerful men, Crassus and Pompey (Caesar married his daughter Julia to Pompey).
These three men worked together for their various political benefits. Crassus died in 53 B.C.E., however, and Caesar's civil war with Pompey led to Pompey's death in 48 B.C.E. With Pompey and Crassus gone, very little stood in Caesar's way of becoming the political dynamo that the dictator Sulla had predicted he would.