Bob_NC is right. Jonsson's book is by far the best and most comprehensive discussion of the details around.
In a nutshell, the date (actually 587 B.C. is probably the correct one, but good arguments exist for both dates because of some apparent ambiguity in the Bible) can be found by working either backwards or forwards from certain astronomically confirmed dates, to the accession year of Nebuchadnezzar (605 B.C.). Then we use several Bible statements that indicate that Nebuchadnezzar destroyed Jerusalem in his 18th or 19th year, which brings us to 587 or 586 B.C.
An example of working backward from an astronomically confirmed date is to use some information from a cuneiform tablet labeled VAT4956. This tablet indicates that in Nebuchadnezzar's 37th year certain astronomical phenomena including a lunar eclipse were observed by the Babylonian scribes. These phenomena, taken together, point to only one year, 568/7 for Nebuchadnezzar's 37th year. Working backwards 19 years gets us to Nebuchadnezzar's 18th year and 587 B.C. Because of uncertainties in the exact dating of Jerusalem's destruction and related events in the Bible, and the fact that a number of scholars appear unaware of information that Jonsson points out, some scholars argue that 586 B.C. fits the data well.
Many other lines of evidence establish the complete chronology of the Neo-Babylonian era, from the 1st king in the dynasty (Nabopolassar; 626-605) to the last (Nabonidus; 556-539). Such evidence includes dated tablets recording everyday events such as business transactions, which are dated by referring to a particular day in a particular month in a particular year of reign of some king. In most cases tablets have been found that are within a few days of the end of a king's reign, and within a few days of the beginning of the next king, which proves that the tablets provide a nearly continuous record of king's reigns. Using this kind of data, it is established that Nebuchadnezzar acceded to the throne in 605 B.C., and from this date scholars derive many other dates in the period. See Jonsson for details.
An example of working forward from an astronomically confirmed date is to use some information recorded in Ptolemy's records. Ptolemy recorded that a lunar eclipse occurred in 621 B.C. in the 5th regnal year of Nabopolassar. This establishes that Nabopolassar's accession year was 626 B.C. Of course, this fits in perfectly with many other records, and helps confirm the standard secular dating for the kings of Babylon from 626 through 539 B.C. Scholars can also go back further, using records of Assyrian kings, to other astronomically confirmed dates which fit perfectly with later dates. Again, see Jonsson for details.
AlanF