The density in the singularity (centre) of the black hole is in theory infinite. But the event horizon (the part where light no longer escapes) is not the singularity, it's simply where the gravity becomes so strong that light can't escape.
Think of it as the sun vs the planets - we're not in the sun, but we still feel its effects. The density of the solar system isn't the same as the density of the sun. This is bad analogy because the same mathematics/physics doesn't apply, but it should help you get the general picture based on your original assumption.
In general, the heavier the black hole, the less dense it is when measured from the event horizon. So in theory, it's possible to have a black hole so heavy that the event horizon contains the entire universe. In fact, the known universe is heavy enough to be a black hole 3 times the known radius of the universe. But as we know from stars that turn into black holes, just because something is heavy enough to be a black hole, doesn't mean it is one yet.