Where do we learn that the deaths of Peter and Paul were during the Neronic persecutions of 64 - 68 AD? What is the source of this information and is it considered valid?
Of the recorded history of Paul's death, Foxe quotes :
Eusebius, Hierom, Maximus, and other authors do but briefly pass it over; so Abdias, (if his book be of any substantial authority,). . . .
Of the sources of the record of Peter's death, he says . . .
Hierom, Egesippus, Eusebius, Abdias, and others, although they do not all precisely agree in the time.
So it would seem that there are limitations in the availability of verifiable historic material.
The first mention is from the Epistle of Clement written circa 96 CE. It mentions that both Peter and Paul were martyred. Though not specifying clearly that it was during Nero's reign, it may be inferred (see footnote 31 below).
Peter, through unrighteous envy, endured not one or two, but numerous labours and when he had at length suffered martyrdom, departed to the place of glory due to him. Owing to envy, Paul also obtained the reward of patient endurance, after being seven times thrown into captivity,28 compelled29 to flee, and stoned. After preaching both in the east and west, he gained the illustrious reputation due to his faith, having taught righteousness to the whole world, and come to the extreme limit of the west,30 and suffered martyrdom under the prefects.31
footnote 31: 31 That is, under Tigellinus and Sabinus, in the last year of the Emperor Nero; but some think Helius and Polycletus are referred to; and others, both here and in the preceding sentence, regard the words as denoting simply the witness borne by Peter and Paul to the truth of the gospel before the rulers of the earth. http://www.ccel.org/ccel/schaff/anf01.ii.ii.v.html
Origen (quoted by Eusebius) declared that Paul was beheaded in Rome under the reign of Nero. Although Clement of Rome writes about the martyrdom of Peter, he does not mention Rome. There does not appear to be any dispute about the time, the place and the manner of Paul’s beheading, but not everyone accepts the Church tradition that Peter was martyred in Rome. Here is a partial quote from an article on the subject:
541 This tradition, that Paul suffered martyrdom in Rome, is early and universal, and disputed by no counter-tradition and may be accepted as the one certain historical fact known about Paul outside of the New Testament accounts. Clement (Ad. Cor. chap. 5) is the first to mention the death of Paul, and seems to imply, though he does not directly state, that his death took place in Rome during the persecution of Nero. Caius (quoted below, §7), a writer of the first quarter of the third century, is another witness to his death in Rome, as is also Dionysius of Corinth (quoted below, §8) of the second century. Origen (quoted by Euseb. III. 1) states that he was martyred in Rome under Nero. Tertullian (at the end of the second century), in his De præscriptione Hær. chap. 36, is still more distinct, recording that Paul was beheaded in Rome. Eusebius and Jerome accept this tradition unhesitatingly, and we may do likewise. As a Roman citizen, we should expect him to meet death by the sword.
542 The tradition that Peter suffered martyrdom in Rome is as old and as universal as that in regard to Paul, but owing to a great amount of falsehood which became mixed with the original tradition by the end of the second century the whole has been rejected as untrue by some modern critics, who go so far as to deny that Peter was ever at Rome. (See especially Lipsius’ Die Quellen der römischen Petrus-Sage, Kiel, 1872; a summary of his view is given by Jackson in the Presbyterian Quarterly and Princeton Review, 1876, p. 265 sq. In Lipsius’ latest work upon this subject, Die Acta Pauli und Petri, 1887, he makes important concessions.) The tradition is, however, too strong to be set aside, and there is absolutely no trace of any conflicting tradition. We may therefore assume it as overwhelmingly probable that Peter was in Rome and suffered martyrdom there. His martyrdom is plainly referred to in John xxi. 10, though the place of it is not given. The first extra-biblical witness to it is Clement of Rome. He also leaves the place of the martyrdom unspecified (Ad Cor. 5), but he evidently assumes the place as well known, and indeed it is impossible that the early Church could have known of the death of Peter and Paul without knowing where they died, and there is in neither case a single opposing tradition.