Can the density of the entire human body be increased at all? And if yes, then what is the maximum amount to which it can be increased to?
There's two ways to increase your density, but both are exorbitantly expensive and probably wouldn't make you bullet proof.
By living on a diet of food made with heavy but stable rare isotopes of the normal atoms, you might be able boost your density by as much as 20%! It would be laughably expensive, and probably wouldn't make you bullet proof though. Then again, I don't suppose anyone's actually tried yet! These heavy elements would include things like using Deuterium in the place of Hydrogen, using Oxygen-18 in the place of Oxygen-16 and so on. As an item of note, these isotopes do exhibit slightly different chemistry from normal, so some minor-to-moderate physiological maladies may result in a real world example.
After that, you could probably boost density by an additional 20%-40% by having the person eat many kilograms of Gold nanoparticles. Eukaryotic (eg. people) cells can soak up relatively huge amounts of these little particles without having problems. Since the particles are mostly absorbed through phagocytosis (the same mechanism our white blood cells use to eat and kill bacteria), the main health risk would be increasing the white blood cells' mass too much relative to the rest of the blood and body. Other cells will absorb them as well, but to a lesser degree.
Again, I don't think these would make you bullet proof, but they would make you heavy! Perhaps a futuristic flexible body-armor would be a more functional alternative? In any case, best of luck with your world.