The word deadline in the sense that we know it today first emerged in early twentieth-century newspaper slang. It’s thought to have originated from a Civil War term (frequently stylized with a hyphen) referring to an actual physical boundary in prisoner-of-war camps that inmates were not allowed to cross, on penalty of death. The term was popularized by its use in the highly-publicized trial of Confederate general Henry Wirz, who infamously established such a line at his squalid camp in Andersonville, Georgia. Later on, apparently people metaphorically extended the word to time limits on the sense that submitting something late is like getting shot. One piece of evidence that would challenge this theory, however, is that there are attestations of the noun deadline being used in the 1920s to refer to a marking on the bed of a printing press. Maybe that’s also related, though – it’s uncertain.
