It seems more likely that Nero did little to persecute the early Christians. The strongest case is that he may have blamed some Christians for starting the Great Fire of Rome, but even here the evidence is inconclusive. Overall, it appears more likely that Nero largely ignored the Christians, as did most other Roman emperors.
The second-century Roman historian, Suetonius, writing of Nero (54-68 CE): "Punishment was meted out to the Christians, a class of men professing a new and wicked superstition." But he does not say what punishment was meted out, nor how often.
Another second-century Roman historian, Tacitus, is shown to have said of the Christians after the Great Fire of Rome, " ...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." However, some scholars see this as a "later, pious interpolation by Christian apologists". Richard Holland (Nero: The Man Behind the Myth) says, "Perhaps the strangest thing is that none of the gospels, nor even Acts of the Apostles, makes the slightest reference to the persecution by Nero of any Christians at Rome, let alone 'a great multitude' of them. "
The apostles Peter and Paul are traditionally said to have been executed during the reign of Nero, but this tradition appears to have begun long after the end of the first century. Acts certainly does not mention these executions, although it can be assumed that it would have done so if the early Christians knew of them.
Whilst the reliability of Tacitus is indeed questioned; there is no solid grounds for doubting his claims and as virtually the only chronicler of Nero's relation to Christianity whose work has survived, he cannot be so easily dismissed. Tacitus, continuing his above quote, goes on to say of the Christians that:
Besides being put to death they were made to serve as objects of amusement; they were covered with wild beasts' skins and torn to death by dogs. Some were crucified, others set on fire to serve to illuminate the night when daylight failed, fastened on crosses, and, when daylight failed, covered by inflammable matter, were set on fire to serve as torches during the night. Or tied to stakes in Nero's gardens while he drove around in his chariot, naked, indulging himself in his midnight revels, gloating over the dying agonies of his victims. The Roman Christians, accused by Nero of setting the city on fire, were massacred in a spectacular fashion on the Vatican Hill.
Which lays out a comprehensive list of persecutions that Nero inflicted upon Christians.
Another point: the Bible indeed makes many allusions indeed to the martyrdom of Christians. The book of Revelation is widely believed to be addressing Rome in its references to Babylon (this was a code of sorts, to prevent early readers from being persecuted); and note just two examples of martyrdom that are found in that book:
Revelation 6:9 And when he had opened the fifth seal, I saw under the altar the souls of them that were slain for the word of God, and for the testimony which they held:
Revelation 20:4 And I saw thrones, and they sat upon them, and judgment was given unto them: and I saw the souls of them that were beheaded for the witness of Jesus, and for the word of God, and which had not worshipped the beast, neither his image, neither had received his mark upon their foreheads, or in their hands; and they lived and reigned with Christ a thousand years.