Answer
It seems that Nero blamed the Christians of Rome for starting the Great Fire. Although he did not start the fire, and it seems that his actions during the emergency were exemplary, he could not avoid rumours that he was involved, or at least that the gods sent the fire to punish him. Tacitus
(Annals 15.44), writing almost a hundred years after the event, says:
To quash the rumour, therefore, Nero substituted other culprits, those commonly known as Christians, who were hated for their disgraceful activities - and he inflicted excruciating punishments on them. Christus, the originator of this name [of Christians], had been executed by the procurator Pontius Pilate during the reign of Tiberius. This deadly superstition, though repressed for a while, was breaking out again, not only in Judaea where the evil originated, but even in the city [Rome], where all atrocious or shameful things flow in from all sides and flourish. So, at first, those who acknowledged it were arrested, and then by their evidence a great multitude were convicted, not so much for the crime of arson as for their hatred of the human race.
All we have from this is a statement that excruciating punishments were inflicted on the Christians, although this would not necessarily involve execution. Suetonius lends some support to Tacitus, "Punishment was meted out to the Christians, a class of men professing a new and wicked superstition."
On the other hand, the French scholar G. Walter discusses the possibility that part of the Tacitus reference is a later 'pious interpolation' by Christian apologists. Warnington (
Nero Reality and Legend) says, "No other writer, Christian or pagan, in the following centuries refers to Nero using the Christians as scapegoats, although Christian tradition knew of Nero as a persecutor ..." Richard Holland (
Nero: The Man Behind the Myth) says, "The Neronian persecution of Christians was an isolated phenomenon, not repeated elsewhere in the Empire at that time."
Thus we have a disputed statement that Nero persecuted the Christians of Rome, but no reliable information as to what form that persecution took. We do not know how the Christians were killed.