If that's the same water your using to wash your teeth I think I'd do more than add some chemicals in the hope of clearing the system. My suggestion:
Turn off main cold feed to tank, run off all water- hot and cold. Examine tank- why is algae forming? Is the tank covered? Is the tank too big, are the draw off points suitably located? Try and establish a reason for this occurence.
Manually clean off the inside of the tank and use copious amounts of water and a sterilising solution like Milton. Flush through the system (filling and draining several times) until all traces of dirt and disinfectant are gone).
When the job is complete monitor your cistern over a few weeks/ months- you should not have a reoccurence of the problem- if you do there is a fault with the installation that's allowing stagnation to occur.