Survivalist, you should know that drinking large amounts of distilled water is bad for you. Since it is free of minerals, it strips them from your system. This is not a worry if you are talking about stashing 2L bottles, but if SHTF and you are living off 55g drums of the stuff it can be hard on your system.
Regarding chlorine and bleach, these are excellent ways to keep things sterile. I'd be leary many forms of pool chlorine, they are not designed to be consumed. Bleach uses sodium hypochlorite as a chlorine source and it's by products are relatively innocuous. Here is a link to the MSDS for household bleach, it shows the ingredients to be 3% sodium hypochlorite and 97% water. Bleach should work great provide you can find the old school non-scented variety.
http://florawww.eeb.uconn.edu/msds/bleach_msds.pdf
Most cities probably use hypochlorite as their chlorine source for drinking water. However if this is a concern to you, the chlorine can be removed by a simple charcoal filter before consumption. Also, it is very important that your containers are clean. Clean water, in a clean container, stored out of the sun provides little food for organisms to consume and thus microbial populations are limited. If heavily chlorinated enough, syrup jugs may work, but would not be ideal. I would suggest finding some cleaner containers.