The first editions of Marlowe's Doctor Faustus were published in 1604 (the "A-Text") and 1616 (the "B-text") and not between 1588 and 1593, as Peter Shor claims.
Based on Gabriel Harvey's marginal notes in A. P. Gasser's Historiarum, it is possible that "Johannes Faustius" (yes, Faustius, not Faustus) was talked about in the halls of Cambridge University, where Marlowe also studied. The oldest surviving book about Faust was the Historia von D. Johann Fausten, dem weitbescheyten Zauberer und Schwarzkünstler ("History of Dr. Johann Faust, the Famous Magician and Necromancer"), published in Frankfurt in 1587. Park Honan thinks that an older Latin tract about Faust may have circulated at Cambridge University, but that document is now lost.
The German book was translated into English as The History of the Damnable Life and Deserved Death of Doctor John Faustus by a certain P. F. The earliest surviving edition is dated 1592, but its title page says it is "newly printed" and "amended", so it may have been a reprint of an older edition published as early as 1587. Marlowe may have read that earlier edition, which is now lost.
Marlowe's Doctor Faustus appears to have been completed by 1589, since it was staged at the Bel Savage around that time. There is also a popular sheet entitled "A ballad of the life and deathe of Doctor Faustus the great Cunngerer", registered on 28 February 1589.
How much of that original play text survived into the editions of 1604 and 1616 is still a matter of debate. Both versions include lines that cannot be Marlowe's.
In summary, Marlowe's Doctor Faustus was probably written between 1587 and early 1589, but first published in 1604, just over a decade after the author's death.
Source: Park Honan: Christopher Marlowe: Poet & Spy. Oxford University Press, 2005, pages 198-200.