Help! New to pools with brown pool dust!

[frusteratedinNJ]

We bought our very first pool this year for the kids, a Summer Escapes 13 inch, 2400 gallon (maybe 2200- can't remember). Anyway, we had an issue about a week ago, where the pool water had a slight greenish tint to it. Had the water tested, got the PH and alkalinity adjusted, shocked it a couple times, added algaecide and what's left is a brown dust in piles at the bottom. The water however is crystal clear. The pool has a filter you replace every 2 weeks. Since this brown dust has appeared, I have to rinse the filter every hour to get all the brown muck off of it, and let the water flow clear in the filter area itself. I do this all day long for 8+ hours, and it's showing no difference. (about a week now been doing this). The fine particles are so small that they run right through the vacuum mesh bag (it's a cheap $15 hose powered vacuum with a mesh bag on top that collects the dirt). Swishing the water around looks like it helps, however hours later, the dust settles again. It never goes away. I read where someone suggested a sock with a rubber band on the filter going back into the pool. I will attempt to try that today but it requires getting into the pool. The space around the hose going into the pool is not that big so I am wondering if I will be able to get a rubber band around it.
So we've had our very first pool a little over a month and I am ready to take it down for the season. I see pictures of people's pools that are so clean and I want my pool like that!! Is there a chemical that will dissolve this brown dust? Help!