Your picture of an expanding balloon is not bad. But it's more like the universe is located on the SURFACE of that balloon. We do not have a "center" of the universe. Of course, this is a rough translation, as the universe is 3D and the balloon surface only 2D. Now you just have to think about a 4D balloon with a 3D surface
So, keeping that picture, it is very difficult to tell what the universe is expanding into, as there is no "into" - at least not for us. A two-dimensional being living on the surface of said balloon would not be able to comprehend the existance or location of an "inside" or "outside" of that balloon. For the same reason, it is not possible to locate the spot of the Big Bang, as it is equally distant from every spot in the universe but not located within that universe any more - we are not able to see the "inside" of that balloon and to the center there.