I really do need to look into the magic that is activated carbon filtration, but I do know it has something to do with the incredible amount of surface area that activated carbon has. Regardless of whether I actually know the science behind it, there is a great link (in not-so-great English) on how to build your own charcoal water filter from just about nothing.
I’ll paraphrase here:
There is a picture at the site that further explains what proportions to use.
The key is that the charcoal you’ll use will come straight from your very own campfire. That’s right, just go through the remnants of your fire (watch out for hot coals!) and pick out the black nuggets that are the charcoal. For such a potentially life-saving device, it’s incredibly easy to make.