Chlorine dude... Brushing and swirling by hand to make sure it's mixed well.
Shocking the pool without a pump isn't easy but doable if you can put in the manual work to mix it. If you add bleach without mixing it well you'll damage the liner.
Have you read pool school? If not, read read read it. Many times.
A test kit will also serve you well because without one, it's next to impossible to shock the pool properly. If you don't shock it properly, it will just re-bloom the algae.
If you have any test results, that will help us help you.