The expression to drop dead, used literally to mean 'to fall down dead', is first known in the mid nineteenth century, and seems to have been popular from then on, since a number of examples have been found thereafter. "I was expecting them momently to drop dead, for both were blind, red with blood, and so exhausted that they frequently fell down" ("Mark Twain," Life on the Mississippi (1883)); "I didn't get brain fever, I did not drop dead either" (Joseph Conrad, Lord Jim (1900)).
Slang use is first known in the early twentieth century, when drop dead! became common as an interjection used to express rejection and contempt, roughly equivalent to go to hell! or the like.
Our expression, drop-dead, as an adjective or adverb meaning 'dazzling(ly); awe-inspiring(ly); astonishing(ly)', is first found about 1970. (The form drop-dead designating the last possible time for something--"what's the drop-dead date for these corrections?"--is earlier, but probably not directly related.) It seems to be frequent in the fashion industry, and many of the examples refer to looks--a drop-dead outfit, a model who's drop-dead beautiful, etc. The equating of death with wonderful things is paralleled by a variety of other slangy expressions; the best recent comparison is to die for, as in "That dress was to die for!"