For a branded Camelbak, their complete cleaning kit comes with a gadget that holds it open for drying. It works really well.
No matter what brand your bladder is, I recommend the cleaning kit just for the drink tube brush. There are other ways to clean the drink tube, but they're a huge PITA. If your bladder is a brand other than Camelbak, you can buy the cleaning kit without the drying frame (which doesn't fit other brands).
Whatever brand you use, the key to drying is to hold the bladder completely open, so the sides don't touch each other anywhere. That's actually the only thing I don't like about my Nathan pack - the mouth is so small - but a couple of things that will go into it, and are light enough to stay put while the bladder dries, are a slim spatula/scraper (turned sideways) or a slim wooden spoon.
Once you've got it held open, you can just pin it by the corners to your indoor clothesline. I don't know how UV-resistant the plastic is, but I'd guess not very, so I wouldn't recommend an outdoor clothesline.