Nero's original name was Lucius Domitius Ahenobarbus, and his father was Cnaeus Domitius Ahenobarbus. The Domitii Ahenobarbi were a distinguished plebeian family with at least seven consulships to their credit (according to Suetonius' Life of Nero), promoted to patrician rank by Octavian around 30 BC.
Cnaeus was the son of another Lucius Domitius Ahenobarbus and Antonia the Elder (sister of Antonia Minor - Claudius, Germanicus and Livilla's mother). Both Antonias were the daughters of Marcus Antonius and Octavia, Octavian's (ie Augustus's) sister, from that period during the Second Triumvirate when Antonius was married to Octavia (from 40 BC until after 37 BC when Antonius went back to Cleopatra).
So Nero's father was related by blood to Augustus, being descended from Augustus's sister. Hence Nero can be said to have had Julian blood. And three emperors (ironically perhaps) were descended from Augustus's rival Marcus Antonius - Gaius, Claudius and Nero.
Interestingly, of Cnaeus's (Nero's father) sisters, one was the mother of Messalina (Claudius's wife), and another was married to Passienus Crispus who divorced her to become Agrippina's second husband.