Laundry bleach has alot of additives in it such as softeners and stuff. However, when my boys had one of those little fill every day pools I used to put a cup of bleach and a 1/2 cup of baking soda in it to keep it clean for the day. I would empty it out each night. I had read on line that this was ok for their skin and to keep bugs like mosquitos out.
You can buy chlorine tabs pretty cheap at places like Odd lots or Big Lots. Depending on the size pool you might spend around $30 -$50 right away but they tabs will last the entire summer.