In a recent interview with rockpapershotgun.com, Markus Persson (better known as Notch), has stated that he would like to add a narrative to Minecraft. In the interview he says that it wouldn't become the main focus; it would simply help "people know what’s (supposed to be) driving their character." Notch goes on to say that a person could ignore the storyline altogether and it wouldn't affect their experience at all.
However, you shouldn't expect to see a storyline implemented anytime soon, considering Notch is planning this for the official 1.0 release. Also mentioned in the interview, Notch plans to add an endgame to Minecraft. When asked if he had any ideas for it, Notch had this to say, "No, heh… Hmm… I probably should think about that."