A comment was deleted that suggested Paul's letters were written after the destruction of the Temple (despite Paul being dead by then!). I was typing out this reply, and now I don't want it to go to waste!
Scholars are (so far as I'm aware) unanimous in dating his letters to about the 50s. Here's what Raymond F. Collins says about the dating of the earliest letter, 1 Thessalonians:
"1 Thess was occasioned by Timothy's report to Paul. It is virtually certain that it was written from Corinth almost immediately after Timothy's arrival there from Thessalonica. The impression given in Acts that the events of Paul's Mission II were moving quickly at this point in his life is confirmed by 1 Thess. Paul writes about being separated from the Thessalonians for a short while (1 Thess 2:17). He frequently recalls his personal presence within the Thessalonian community (the "recall motif," 2:1). It would seem, then, that only a relatively short time, probably a few months, intervened between Paul's evangelization of Thessalonica and his writing of 1 Thess. Most probably the letter was written in AD 50 (B. Schwank, A. Suhl), but some scholars continue to date it in AD 51." (NJBC, 773)
The dating of Paul's life is established using extrabiblical sources. Here is Joseph Fitzmyer:
"In Paul's own letters the only incident that can be dated extrabiblically is his Damascus escape (2 Cor 11:32-33): the ethnarch of King Aretas closed off the city to take Paul captive, but he escaped by being let down in a basket through a window in the city (cf. Acts 9:24-25). That occurred at the end of Paul's three years in Damascus (Gal 1:17c-18). Since Damascus was apparently under Roman rule until Tiberius's death (March 16, AD 37; cf. Josephus, Ant. 18.5.3) and the Nabatean Aretas IV Philopatris (9 BC-AD 39) was given control over it by the emperor Gaius Caligula, Paul's escape must have occured between AD 37 and 39, probably in AD 39 (see PW 2/1 [1895] 674). Paul's conversion was about three years earlier, probably in AD 36." (NJBC, 1330)
Fitzmyer then goes into detail about five additional extrabiblical events that help date Paul's career.