The fact that the fire evidently rekindled and that there were two conflagrations in separate parts of the city fed rumors of arson. Nobody wanted to believe that the valiant Vigili, the professional firefighters, aided by denizen volunteers, had failed in initial efforts to extinguish the blaze or that wind-borne embers could land on tinder-box roofs across town. No, it was just too easy to believe that Nero, already despised by a segment of the population, might start a fire for "urban renewal" purposes. Nero's subsequent acquisition of large burnt out areas near the western end of the Forum for construction of his Domus Aurea fed the rumors that his agents at least stoked the blaze if they didn't start it.
When the fire started, Nero was at Antium (Anzio) on the coast south of Rome, but he did rush back to the city as soon as possible and lead the fire-fighting and recovery effort. Even his enemies acknowledged that he opened his palace to those made homeless by the fire and spent personal funds to feed and house many others. His greatest prosecutor, the historian Suetonius (who was born the year after the fire) credited him with a new post-fire building code that required fire-fighting platforms on wooden structures.
But Suetonius was adamant that Nero had instigated the fire in the first place. Tacitus was slightly more cautious, but Dio Casius again stated it as fact. Modern historians disagree for several reasons. First, the "urban renewal" motive is flimsy: he could just as easily sequestered property without a fire. Second, all reports said that there was a bright full moon, so it wasn't a good night for arson. Third, he and his government went to great lengths with disaster relief in the aftermath. At his instigation, the government paid for reconstruction of the city's public buildings, paid bonuses to individuals who rapidly rebuilt their own houses, and enacted the stricter planning and building codes mentioned above: wider, straighter streets, firefighting facilities, water distribution, etc. In addition, destruction of the city would be completely inconsistent with Nero's obsession with his image with the people (as opposed to the aristocratic powers) of Rome. Admittedly, the latter arguments can be turned around, but the first is pretty ironclad: why go to the expense of fire and rebuilding if you already have the power to do and take what you want.
And Nero did have that power. As unpopular as he might have been with the aristos, he had the people on his side, at least until the embers cooled. Only after the post-fire rumors circulated did he start to lose the support of the populace. To counter the efforts of enemy rumor-mongers, Nero chose an easy target -- Christians.
The Christians, popularly believed to be the most troublesome segment of Rome's Jewish minority, were already disliked if not actively hated. They were different and insular, their religious services were private, and what little was "known" about their beliefs was disgusting even to the bloodthirsty Romans. This is what Romans had heard: the infant god of the Christians grew up was eventually eaten, "body and blood" (the Christians said) in what the Romans considered a cannibalistic ritual. And this was something which the Christians reenacted often. Why, they were worse than those horrible ancient Carthaginians, who burned their babies but at least didn't eat them. Any Roman child that went missing led to inevitable conjecture.
Nero blamed the Christians for the fire, collected a few, and had them executed. This show soon became popular in the palace and with the mob, and the first persecution was well under way. Saints Peter and Paul were said to have been among the first to be executed. But the killing of the Christians did not quell suspicions of the emperor. Undoubtedly, his political enemies, of whom there were more and more, encouraged the belief that the Emperor was the flagrant. And this, along with other later Neronian excesses led to the final scene of Nero, in disgrace and condemned by the Senate, fleeing through the city and eventually committing suicide four years after the fire.
But did he fiddle? The obvious answer is that the fiddle wasn't yet invented -- there wouldn't be a true fiddle or violin until the Renaissance. But he did have a lyre, and he was said to have written an ode and presumably its accompaniment comparing the Roman fire of 64 AD with the destruction of Troy. Nero might have claimed that he was honoring Rome by comparing its tragedy with the destruction of the ancestral home of Romulus and Remus and therefore of his own Julio-Claudian line (his mom, who he had already bumped off, was a direct descendant of Julius Caesar, who claimed to be the scion of that Iulus, the son that Aeneas had rescued when he left burning Troy.) His later enemies, however, apologists all for the succeeding dynasty's assumption of power, projected the ode (and the lyrical fiddling) as another example of the frivolous Emperor, playing at the arts instead of caring for his people. Ancient legend says he watched the city burn (and fiddled) from a vantage point in a tower in the gardens of Maecenas on the Esquiline Hill. Later legends, and dozens of generations of tour guides, put the fiddling in the Torre delle Milizie, which still juts above the slopes of the Quirinale Hill. But that tower, in its present form, dates from an early 13th century aggrandizement by the Caetani family of an ancient Republican wall tower.
Suetonius on the fire (scroll down to section XXXVIII): http://www.fordham.edu/halsall/ancient/suet-nero-rolfe.html
Dio Cassius on the fire: http://www.fordham.edu/halsall/ancient/diocassius-nero1.html
"Quo Vadis" is a book about the time of Nero, featuring the fire. Text is on line at ftp://ftp.ibiblio.org/pub/docs/books/gutenberg/etext01/quvds10.txt
Tacitus on the Neronian persecution of Christians: http://www.wsu.edu:8080/~wldciv/world_civ_reader/world_civ_reader_1/tacitus.html
Violin History: http://www.theviolin.net/en/section/1/
Torre della Milizie: http://www.italycyberguide.com/Geography/cities/rome2000/E140b.htm
P.S.: Nero's quarrel against the old power structure was only the latest episode of a centuries old struggle between the self-styled "optimates" (the "best people") and the "populares" (everyone else). It went back at least as far as the Gracchi brothers (ca. 120 BC) -- popular reformers killed off by the Optimates. It continued through the struggles between Marius and Sulla, through the Social War that spread citizenship throughout the peninsula, through the Pompey/Caesar civil war, through Caesar's seizure of power, ostensibly on behalf of the Populares, and through his death at the hands of the Senatorial Optimates, who claimed to seek the restoration of the Republic but were really trying to keep the old families in their dominant position. Augustus kept the dispute under control throughout his long reign, the Pax Romana (until 14 AD), but he began it with a long civil war (44 BC - 30 BC) fought, essentially, between the two factions. After Augustus, the Pax Romana quickly crumbled in the hands of his inept dynastic successors, the last of which was Nero. Nero was woefully inadequate, a spoiled teenager, with no real training in authority, no experience, a warped sense of priorities and his own position, and few if any scruples. He was an easy mark for the Optimates.