ANSWER: The Bible does not tell us the exact time or manner of the apostle Paul's death, and secular history has yet to provide us with any definitive information. However, evidence highly suggests the apostle Paul's death occurred after his fifth missionary journey ended in 67 A.D. Paul was likely beheaded by the Romans, under Emperor Nero, sometime around May or June of 68 A.D. Nero himself died by suicide on June 9th of the same year.
Christian tradition also has Paul being behead in Rome around the mid 60s A.D. during the reign of Nero. Most Bible dictionaries and some commentaries can give us details on the traditions surrounding Paul's death.
"Concerning the time, place, and manner of his death, we have little certainty. It is commonly believed that, when a general persecution was raised against the Christians by Nero, about A.D. 64, under pretence that they had set Rome on fire, both St. Paul and St. Peter then sealed the truth with their blood; the latter being crucified with his head downward; the former being beheaded, either in A.D. 64 or 65, and buried in the Via Ostiensis. EUSEBIUS, Hist, Eccles. lib. ii. cap. 25, intimates that the tombs of these two apostles, with their inscriptions, were extant in his time; and quotes as his authority a holy man of the name of Caius, who wrote against the sect of the Cataphrygians, who has asserted this, as from his personal knowledge. See Eusebius, by Reading, vol. i. p. 83; and see Dr. Lardner, in his life of this apostle, who examines this account with his usual perspicuity and candor.
"Other writers have been more particular concerning his death: they say that it was not by the command of Nero that he was martyred, but by that of the prefects of the city, Nero being then absent; that he was beheaded at Aquae Salviae, about three miles from Rome, on February 22; that he could not be crucified, as Peter was, because he was a freeman of the city of Rome.
"But there is great uncertainty on these subjects, so that we cannot positively rely on any account that even the ancients have transmitted to us concerning the death of this apostle; and much less on the accounts given by the moderns; and least of all on those which are to be found in the Martyrologists. Whether Paul ever returned after this to Rome has not yet been satisfactorily proved. It is probable that he did, and suffered death there, as stated above; but still we have no certainty" (Commentary on the Bible by Adam Clarke, commenting on Acts 28:31).
The following excerpt is from The International Standard Bible Encyclopedia, First Edition, article Paul the Apostle.
"When Paul writes again to Timothy he has had a winter in prison, and has suffered greatly from the cold and does not wish to spend another winter in the Mamertine (probably) prison (2Timothy 4:13, 21). We do not know what the charges now are. They may have been connected with the burning of Rome. There were plenty of informers eager to win favor with Nero. Proof was not now necessary.
"Christianity is no longer a religion under the shelter of Judaism. It is now a crime to be a Christian. It is dangerous to be seen with Paul now, and he feels the desertion keenly (2Timothy 1:15ff; 4:10). Only Luke, the beloved physician, is with Paul (2Timothy 4:11), and such faithful ones as live in Rome still in hiding (2Timothy 4:21).
"Paul hopes that Timothy may come and bring Mark also (2Timothy 4:11). Apparently Timothy did come and was put into prison (Hebrews 13:23). Paul is not afraid. He knows that he will die. He has escaped the mouth of the lion (2Timothy 4:17), but he will die (2Timothy 4:18). The Lord Jesus stood by him, perhaps in visible presence (2Timothy 4:17)."The tradition is, for now Paul fails us, that Paul, as a Roman citizen, was beheaded on the Ostian Road just outside of Rome. Nero died June, 68 A.D., so that Paul was executed before that date, perhaps in the late spring of that year (or 67). Perhaps Luke and Timothy were with him. It is fitting, as Findlay suggests, to let Paul's words in 2Timothy 4:6-8 serve for his own epitaph. He was ready to go to be with Jesus, as he had long wished to be (Philippians 1:23)"
Lastly, the following quote regarding the death of Paul was taken from the Smith's Bible Dictionary by Dr. William Smith, article entitled Paul.
"This epistle, [2Timothy] surely no unworthy utterance at such an age and in such an hour even of a St. Paul, brings us, it may well be presumed, close to the end of his life. For what remains, we have the concurrent testimony of ecclesiastical antiquity, that he was beheaded at Rome, about the same time that St. Peter was crucified there. The earliest allusion to the death of St. Paul is in that sentence from Clemens Romanus, . . . which just fails of giving us any particulars upon which we can conclusively rely. The next authorities are those quoted by Eusebius in his H. E. ii. 25. Dionysius, bishop of Corinth (A. D. 170), says that Peter and Paul went to Italy and taught there together, and suffered martyrdom about the same time. This, like most of the statements relating to the death of St. Paul, is mixed up with the tradition, with which we are not here immediately concerned, of the work of St. Peter at Rome.
"Caius of Rome, supposed to be writing within the 2nd century, names the grave of St. Peter on the Vatican, and that of St. Paul on the Ostian way. Eusebius himself entirely adopts the tradition that St. Paul was beheaded under Nero at Rome. Amongst other early testimonies, we have that of Tertullian, who says (De Praescr. Haeret. 36) that at Rome:
"Petrus passioni Dominicae adaequatur, Paulus Johannis [the Baptist] exitu coronatur;"
"and that of Jerome (Cat. Sc. Paulus):
"Hic ergo 14 to Neronis anno (eodem die quo Petrus) Romae pro Christo capite truncatus sepultusque est, in via Ostiensi."
"It would be useless to enumerate further testimonies of what is undisputed."