If you have no chlorine in the pool, the pool will turn green. Plain and simple. And, it doesn't take long before algae starts.

Bleach is 8.25% sodium hypochlorite. Pool stores also sell liquid chlorine. It is either 10 or 12.5% sodium hypochlorite. The same stuff only more concentrated. People do not hesitate to add pool store chlorine but when they hear the word 'bleach,' they get scared. If you wouldn't be afraid to add pool store chlorine, then you shouldn't be afraid to add bleach. (In all the years I have had a pool, I have never used anything but bleach for my source of chlorine.)

Those gallons of bleach aren't actually gallons but rather they are 121-oz jugs. In a 23K pool, adding two of those jugs is only going to take your chlorine level up to a little less than 7ppm which is not all that high. Nothing to worry about at all.

Test the water again this evening and repost with the numbers. It is likely that you will need to add more bleach this evening as well. And, go ahead and order a good kit ASAP using the link Ben gave you so you'll get it as soon as possible.