For more details about how bleach (hypochlorous acid) kills bacteria, see
this post. It refers to an article and scientific paper where the mechanisms are described.
What Jason described with respect to disruption of the cellular wall to create leakage is how common algaecides work (e.g. quaternary ammonium compounds including Polyquat) and more details for that are in
this post. Hypochlorous acid, however, looks very much like a water molecule so tends to get passed through cell membranes and does most of its damage inside the cell, mostly reacting with proteins that have amine (nitrogen) or sulfur groups and this causes the proteins to unfold and then aggregate. Cells need proteins to function properly to survive so disruption of proteins prevents cell reproduction and the cell essentially dies.
There is also a much slower attack on cell walls by hypochlorous acid but such attack shows up mostly for pathogens with thicker cell walls including protozoan oocysts that take far longer to inactivate. The chlorine reaction is much slower because such cell walls are composed mostly of sugars that do not contain nitrogen or sulfur groups subject to faster attack by chlorine (the peptidoglycan, lipopolysaccharaides, lipoproteins, and chitin have their most common nitrogen component in an amide group that reacts very slowly with chlorine). Also, for protozoan oocysts, they do not pass much water through their shells and therefore not much chlorine. Such cysts are activated after ingestion by the low pH in the stomach followed by pancreatic proteases in the small intestine (see
this paper for more info).
Many water utility districts have switched the chemical used as the residual in the mains from chlorine to monochloramine. The reason is two-fold. First is that they wanted to reduce the amount of disinfection by-products. Second is that they wanted the disinfectant residual to last longer. Monochloramine reacts more slowly (so lasts longer) than chlorine and does not produce as many disinfection by-products. It will kill many bacteria faster than they can grow, but it will not generally kill them fast enough to prevent person-to-person transmission of disease nor will it inactivate viruses or protozoan oocysts that may be introduced by bathers. What's good for drinking water is not so good for swimming pools.
