Go solar. Fill a black plastic garbage bag (no holes!) with snow, and place it in the direct sun; wait two to three hours for it to melt, then boil the water to purify it. The same trick works with black water bottles. Only have clear bottles? No problem: Wrap them in black duct tape (gray might get the job done, too, but not as efficiently); plus, you then have extra duct tape handy.
In a covered pot, warm up about an inch of water, then add snow by the handful as it melts. Never put a pot filled with only snow atop a stove flame. Because snow is an insulator—not a heat conductor—you’ll burn your pot before you melt the snow. Also, don’t wait until the water boils before adding more snow—this just wastes fuel. Hint: Clever campers fill a second pot with snow, cover it, and set it atop the first to capitalize on rising heat.