Chlorine...how much are you using and how long are you letting it in the tank?
With a rather strong concentration, you can tow it around a bit to slosh around and clean up the algae, like some do with ice in their black tanks.
If you really have a lot, it is possible you have some sort of organic debris in the tank. Is the top or some other spot accessible? Make sure no dirt/moss got into the tank.
Maybe it was filled from a pond at one time to get it started?
Another thing to consider is using white vinegar instead of bleach.
Soap (phosphorus) can feed and encourage algae in creeks, likely the same in the tank. Have you ever added any soap to the tank?
Hope this helps.