How do you correctly change water in a freshwater fish tank with algae?

I have now killed two beta fish after changing the water 1/2 way after algae growth (it's a 3 gallon tank with a filter). Both time it resulted in the fishes death within a week after the water change. What is the correct way to clean a freshwater fish tank that has algae growth? I know not to change all the water because of the chemicals and pH level must remain in the tank. I usually would change out the filter every month, add water after water loss from perspiration, change only 1/2 of the water in the tank every month as well as rinse out the tank with water only before replacing old water. I use a chlorine remover for all water changes and I do NOT use soaps or other cleaning chemicals (obviously). I really need your help. I want to get another fish but not until I know how to make sure it lives for longer than 9 months.