Skin bleaching doesn't turn skin pure white, but it is achievable to get something close to it. All it takes is a stretching of the truth, much like a Microwave Emitter would never exist in that capacity, and would fry every person and explode all metal within its range. They had to stretch the truth (quite a lot, if you think about what I just mentioned) to make it
seem plausible. The atmosphere of the film does the rest. When the film treats it seriously, it's easier for an audience member to take it seriously.
Oh, and about that picture:
This is someone with vitiligo. All it would take is a
stretching of the truth to get an audience to believe that a skintone like that was the result of a chemical. They don't even need to go into what chemicals, because it would really only be alluded to vaguely, with comments like "Whatever doesn't kill you simply makes you stranger."
Now, of course, compromises would have to be made to adapt it into Nolan's world. I'd have liked to have seen the Joker where he was permawhite, and had the chealsea grin, and added the green hair-dye, and red/black makeup. That way, you get the grittier visual of Nolan's Joker, and the concepts behind permawhite (I'll explain those below).
By the way, if you'd like to see a more realistic interpretation of permawhite, more fully than you see in the picture above, have a look at this:
It's mean to be a corpse's hand in that picture, which I think fits the Joker perfectly: the complextion of a corpse. The skin isn't pure white, either; it's vainy and uneven, a bit darker and greyer in some places. If the Joker were to be permawhite in Nolan's world,
that^ would probably be how it would look.