How much opposition to the Emperor Nero was there during his reign?

A lot at the end. When he first took the throne after the death of Tiberius (murdered on his deathbed by Nero's mother, Agrippina the Younger), Nero was enormously popular. But his extravagance and bizarre behavior alienated the patrician class (Nero murdered his mother among other things) and at the end of his rule he was weakened by military revolts. Facing certain execution, Nero committed suicide, ending the Julio-Claudian dynasty. His death was followed by a period of anarchy and civil war, during which there was a succession of short-lived emperors. Eventually, General Vespasian became emperor and founded a new dynasty. Vespasian is remembered as one of Rome's "good emperors."