We don't use the external pump. We put the chlorine water into the fresh water hose which we've attached to the rig. The other end gets attached to a water faucet and we turn on the water to fill it. We also let it sit for 12 hours before draining the system.
I use the same method as Reverse_snowbird to get the bleach in the fresh water tank. I have a gravity feed port. Use 1/4 cup of regular bleach for every 15 gallons of water.