We are largely dependent on non-biblical sources for the records of the martyrdoms of twelve. This is not surprising when we consider that most of the NT books were written before 70 AD.
Acts 12:2 records the death of James, son of Zebedee, who was executed by Herod.
In 2 Tim 4:6 Paul refers to his impending doom, and in John 21:18-19 Jesus predicts the death of Peter. These deaths are recorded in Clement’s letter to the Corinthian church, probably written in the late first century:
“We should set before our eyes the good apostles. There is Peter, who because of unjust jealousy bore up under hardships not just once or twice, but many times; and having thus borne his witness he went to the place of glory that he deserved. Because of jealousy and strife Paul pointed the way to the prize for endurance. Seven times he bore chains; he was sent into exile and stoned; he served as a herald in both the East and the West; and he received the noble reputation for his faith. He taught righteousness to the whole world, and came to the limits of the West, bearing his witnsss before the rulers. And so he was set free from this world and transported up to the holy place, having become the greatest example of endurance” (First Clement 5:3-7)
For the actual method of their death — beheading (Paul) and crucifixion (Peter) — we are dependent on later historians like Eusebius (HE 2.25.5), though he refers to earlier writings.
As to the other apostles we have less reliable information. According to tradition, the apostles dispersed and went to preach the gospel to various places, but the historical veracity of these reports has been questioned. By tradition, John lived till old age in Epheseus and was the longest lived of the apostles, though it has been suggested that this John was not John the son of Zebedee but another disciple known as John the Elder.