Well, step number one is to give us your test results...hopefully from your own drops-based kit. Avoid test stips, but you probably already know this.

Looking at the water in a cup is not a good comparison. You're looking through 4" of water. It's got to be really funky to not be able to see through that. Looking through 5' of it is where you see the funk.

You need to get your Cl up to the 15 ppm range. You didn't mention the size of your pool, so I can't give you a starting number. However, I can tell you that the only thing that is going to solve your problem is Cl, and lots of it. Do you have a recirc setting on your pump? That may help getting the Cl distributed about. Once the algae is killed off, it won't plug your filter as fast. Sounds odd, but it's true.

Worst case you can dilute the bleach 50% and cast it out in a bucket to disperse it over the surface. That way you shouldn't damage your liner, and you can start killing off the algae so you can clear the pool.

Michael