In addition to chlorine, use a copper based algaecide. They are expensive but actually work. Follow the initial directions for start up and maintenance for the first summer. Then cut back to 1/2 the maintenance dose.
A fancy pump won’t help much unless your piping or filter is under size. Do you have gages that tell you the suction and filter pressures?
There is no magic formula for how long to run the pump. Cut back until it looks like you need more. Add a third. You can cut way back in the winter.
