How did the Apostles die?

Judas, of course, hung himself after betraying Jesus

James is the only other Apostle whose death is recorded in Scripture, in Acts 12:1-2

It was about this time that King Herod arrested some who belonged to the church, intending to persecute them. 2 He had James, the brother of John, put to death with the sword.

The rest of the Apostles deaths are recorded outside of Scripture, and there is some uncertainty and discrepancy.

Andrew - The details are unclear, but the Catholic Encyclopedia lists this.

It is generally agreed that he was crucified by order of the Roman Governor, Aegeas or Aegeates, at Patrae in Achaia, and that he was bound, not nailed, to the cross, in order to prolong his sufferings. The cross on which he suffered is commonly held to have been the decussate cross, now known as St. Andrew's, though the evidence for this view seems to be no older than the fourteenth century. His martyrdom took place during the reign of Nero, on 30 November, A.D. 60); and both the Latin and Greek Churches keep 30 November as his feast.

Bartholomew - Also from the Catholic Encyclopedia

The manner of his death, said to have occurred at Albanopolis in Armenia, is equally uncertain; according to some, he was beheaded, according to others, flayed alive and crucified, head downward, by order of Astyages, for having converted his brother, Polymius, King of Armenia. On account of this latter legend, he is often represented in art (e.g. in Michelangelo's Last Judgment) as flayed and holding in his hand his own skin. His relics are thought by some to be preserved in the church of St. Bartholomew-in-the-Island, at Rome. His feast is celebrated on 24 August. An apocryphal gospel of Bartholomew existed in the early ages.

James, the son of Alpheus is listed in the Orthodox Wiki with the following:

St James finished his apostolic work in the Egyptian city of Ostrachina, where he was crucified by the pagans.

while Foxes Book of Martyrs says this:

At the age of ninety-four he was beat and stoned by the Jews; and finally had his brains dashed out with a fuller’s club.

Peter - from the Catholic Encyclopedia:

It is an indisputably established historical fact that St. Peter laboured in Rome during the last portion of his life, and there ended his earthly course by martyrdom. As to the duration of his Apostolic activity in the Roman capital, the continuity or otherwise of his residence there, the details and success of his labours, and the chronology of his arrival and death, all these questions are uncertain, and can be solved only on hypotheses more or less well-founded. The essential fact is that Peter died at Rome: this constitutes the historical foundation of the claim of the Bishops of Rome to the Apostolic Primacy of Peter.

followed by several accounts, too numerous to list here.

Thomas - from the Orthodox Wiki

Preaching the Gospel earned the holy Apostle Thomas a martyr's death. For having converted the wife and son of the prefect of the Indian city of Meliapur (Melipur), the holy apostle was locked up in prison, suffered torture, and finally, pierced with five spears, he departed to the Lord. Part of the relics of the holy Apostle Thomas are in India, in Hungary and on Mt. Athos.

The Catholic Encyclopedia isn't so sure...

On the other hand, though the tradition that St. Thomas preached in "India" was widely spread in both East and West and is to be found in such writers as Ephraem Syrus, Ambrose, Paulinus, Jerome, and, later Gregory of Tours and others, still it is difficult to discover any adequate support for the long-accepted belief that St. Thomas pushed his missionary journeys as far south as Mylapore, not far from Madras, and there suffered martyrdom.

Philip - Not much information exists on his death. What is given is always prefaced as "rumors". Some accounts say hostile Jews had him tortured and then crucified upside down. Some sources have him being stoned. The honest answer is "we just don't know".

Simon - From the Catholic Encyclopedia:

The Abyssinians accordingly relate that he suffered crucifixion as the Bishop of Jerusalem, after he had preached the Gospel in Samaria. Where he actually preached the Gospel is uncertain. Almost all the lands of the then known world, even as far as Britain, have been mentioned; according to the Greeks, he preached on the Black Sea, in Egypt, Northern Africa, and Britain, while, according to the Latin "Passio Simonis et Judae" — the author of which was (Lipsius maintains) sufficiently familiar with the history of the Parthian Empire in the first century — Simon laboured in Persia, and was there martyred at Suanir. However, Suanir is probably to be sought in Colchis. According to Moses of Chorene, Simon met his death in Weriosphora in Iberia; according to the Georgians, he preached in Colchis. His place of burial is unknown.

Judas Thaddeus - From the orthodox Wiki:

He was one of the Twelve Apostles (not to be confused with the Thaddeus of the Seventy Apostles) and after the Ascension he preached the Gospel in Judea, Samaria, Galilee, Idumea, Syria, Arabia, Mesopotamia, and Armenia. While preaching in the area around Ararat he was captured by pagans, crucified and killed by being shot with arrows.

John was spared a violent death.

Matthias was sentenced to death and stoned.

Paul According to the Orthodox Wiki, Paul was martyred with the Apostle Peter under Nero by beheading.