What chemicals kill algae in pools?

Chlorine - you can buy "pool shock" at Walmart, pool specialty stores, or maybe even Menards. After a couple days of "shocking" a pool, though, any algae with die and turn brown, meaning you'll probably have to scrub the walls and floors of your liner and filter out the dead stuff.

Also, my understanding is that bleach does have the same effect.

Depending on your pool size, I'd say you should use at least 1/2 a gallon of one of these solutions every day, pouring it around the perimeter. Like I said, after a couple days the pool will become brown. If you have an inflatable pool, use much less; you'll probably have to use more bleach than chlorine/shock.