Nero's early reign (when he was under the firm tutelage of Seneca and Burrus) seem pretty good and he followed Augustus' model. Post 59 CE he had removed his former advisers and ran wild. He seems to have been someone given absolute power at a very young age and let it go to his head. My prof was of the opinion that his persecutions against the Christians certainly coloured the transmission of his reign by early Christian sources (in addition to Suetonius' and Tacitus' stern disapproval).

As for his contemporaries, we have records that he was very popular with the everyday people of Rome and he was spoken of positively in Greece into the 2nd C CE. The number of false Nero's suggest he was popular or represented something people were willing to embrace. Policies like proposing to cancel all indirect taxes (he was dissuaded by his advisers as this would wreck the state's finances) reflect this populist touch.

However he was absolutely detested by the senate as he slipped back into an authoritarian mode which quickly put him in their black book. Generally speaking the 'good' emperors were those who treated the senate with respect and listened to their counsels (regardless of whether he took it seriously or not). His love of performance was a horrifying breach of all social norms and etiquette - the elites simply did not put themselves on display in such a manner. It's the modern equivalent of the head of state showing up and performing in 'Jackass'.

As for the rebellion, I'm not sure if we have any contemporary (or near) primary sources for that bit - Tacitus breaks off in 67 and doesn't pick up again till 69 CE. I can only surmise that Vindex's rebellion of 68 was influenced by the forced suicide of Corbulo in 67 CE (attributed to jealousy by Nero). Once that cat was out of the bag, the other generals with armies must have weighed out the factors and decided that his disgraceful behaviour meant he didn't deserve to rule the empire.

Nero (as stated by /u/Agrippa911) was wildly popular when he first came to power. Then he got power hungry. While the "Nero fiddled while Rome burned" saying is false, there is probably a kernel of truth. There were rumors at the time that Nero started the Great Fire of Rome, especially since it cleared up prime land for his Golden Palace, but most modern historians (and Tacitus) believe that's unlikely. I personally am of the opinion that the fire was an accident. However, he certainly profited from the fire. He built the Golden Palace and blamed the Christians.

To me, the most telling sign that he fell out of favor was that he was not deified. Most emperors were, and all of the Julio-Claudians except Caligula and Nero were. When Vespasian took power, the Lex Quae Imperio Vespasiani ("Law about the power of Vespasian," basically the grant that made him emperor) makes reference to Vespasian being granted the powers of "deified Augustus, for Tiberius Julius Caesar Augustus [Tiberius], for Tiberius Claudius Caesar Augustus Germanicus [Claudius]," which rather pointedly leaves out Caligula, Nero, Otho, Vitellius and Galba. The last three were emperor for less than a year each, so the fact that they were not deified or considered important is not surprising. However, Caligula and Nero both had longer reigns and were Julio-Claudians, which makes the lack of deification unusual.