If cosmic inflation is the correct explanation of the history of our universe, then it is possible that outside our visible patch (or Hubble volume) of the universe, the cosmic evolution (expansion rate, initial conditions, physical constants, etc.) could be different. These possibilities are included in what are sometimes called Level I and II multiverse hypotheses. An observer in any individual Hubble volume of the total universe would see his visible universe as a particular cosmological model, but would be unable to communicate his measurements to an observer in a different patch.
However, the expanding universe refers to the expansion of spacetime, not just to the movement of galaxies and stars. Spacetime itself is changing, it's not just matter flying around as if an explosion had occurred. That picture is an inaccurate one used to try to illustrate some concepts to the layperson. However, it is misleading if one tries to attach too much physical significance to it.
We don't really know what the initial state of our universe was, though there are various hypotheses. If we try to follow your suggestion that it was a large black hole, we would still need to explain why it exploded. Large black holes are fairly stable objects. They give off very low temperature Hawking radiation and would be expected to survive for a very long time (think billions of years). After they've radiated off most of their mass (we're now talking several Planck mass-sized black holes, not universe-sized ones), it's expected that they become unstable, since they are now very hot and burning rapidly. The end state of this decay is also an unsolved problem in quantum gravity, but it doesn't directly shed light on the big bang.