Plain, unscented laundry bleach and liquid pool chlorine are one and the same except for the strength. Both are sdoium hypochlorite. There are several kinds of chloirine that are suitable for a pool but they fall into two major groups, stabilized and unstabilzied.
The Stabilized chlorines are trichlor tablets and dichlor granules (which is sometimes sold as 'shock'). Both of these will cause your CYA (stabilizer) levels to rise as you use them and eventually lead to an over stabilized pool with all the problems that brings. With a cartridge filter this will happen in just a few months. It will take a bit longer with a sand or DE filter that is cleaned by backwashing bit it will still happen in most cases. The reason that many people use trichlor tablets in a feeder or floater is because trichlor is a very slow dissolving form of chlorine and it is the only form of chlorine that works with a feeder or floater. It's a esy way to chlrinate a pool until the stabilzier becomes too high.
The unstabilized chlorines are sodium hypochlorite (bleach and liquid pool chlorine), calcium hypochlorite (usually sold as powdered 'shock' but also sold as chlorinating granules) amd lithium hypochlorite (also sold as 'shock'). They do not contain CYA so they will not cause overstabilization.
Calcium Hypochlorite (cal hypo) wil cause your calcium levels to rise since it adds calcium to the water. This might or might not be a problem. It depends on how high your calcium levels are in your pool and fill water.
Lithium hypochlorite's only disadvatage is that is it the most expensive form of chlorine you can buy. If you can afford it go for it! It has no major side effects and it is convenient because it's a fast dissolving granule.
Sodium hypochlorite (bleach) also has no major disadvantages and is a very economical form of chlorine, which is why it is the one we recommend.
When using unstabilized chlorine in any of it's forms you do need to add chlorine to the pool daily but we are talking about 5 munites worth of daily testing and maintenance so it's not that much work and the rewards are a trouble free pool.
Hope this info is useful.