alum will clear up turbid / dingy water for sure. It will also significantly drop the pH of the water (making it more acidic). This could irritate your stomach.
Also, not sure of the FDA status of your alum source. Probably better options for your drinking water.
I've used alum to help settle fine particulates on desert river trips. I let a bucket of river water settle overnight with some alum and then treated it the following morning.
For a strainer, I've used coffee filters, layers of paper towels in a strainer, and a tshirt.
River runners use it all time on SW rivers, like the Grand Canyon that have high sediment. We used it on a September trip a few years back, worked great. Down River Equipment sells another product called water wizard, I haven't used it but have heard it works well. If you search Mountain Buzz I'm sure you could find some threads on it that talk about its use.