Yes, when folks say put bleach, they're talking about bleach like Clorox. It's cheap, but the bottles are bulky, and you have to poor it slowly at one of the return jets so it gets mixed quickly and doesn't splash on you or your clothes.

the reason the chlorine is going down to zero is because it's getting used up--probably getting rid of the dead algae. Just keep it at shock level until it meets those 3 criteria above. Test every evening and morning, and eventually, it will start holding its level over night. The crystal clear water is a really good sign; you're definitely progressing.