Liquid pool chlorine and common household bleach are the same thing, the only difference is strength 1 to 8% Chlorine for home bleach 1% is the el-cheapo no name stuff, 8% is the new concentrated bleach, and common regular bleach is 5-6%. Liquid pool Chlorine is typically 12-15%, one thing to be aware of is that bleach looses its potency (Chlorine off gases over time), the higher the percentage of Chlorine the shorter the shelf life.
Other dry forms of pool Chlorine are bonded with other types of chemicals, some of which are not good to drink on a regular basis, specifically those sold as Stabilized chlorine which helps keep chlorine in pools longer and prevents sunlight from breaking it down as fast.
Of the common dry chlorine products, probably the safest to put in drinking water would be Calcium-Hypo, since it breaks down mostly to Calcium and Chlorine, note I said safest, not safe.