Sara,
Alum, like all aluminum salts, works to clarify water by coagulation. It it classed as a metal salt inorganic coagulant that is most effective (least solubility) at a pH of 6.5. Above or below that, the solubility increases, and the effectiveness drops off.
While it is a coagulant, it requires quite a bit of settling time to separate, and makes a fluffy floc particle that easily breaks apart if you try to scoop it out. The best way to remove it will be to let it settle (turn off the circulation) and suck it off the bottom (it should sink in time). Alum is not in itself an algecide like chlorine or commercial algecides. It would only kill algae from settling and coagulation. It will work on most solids by trapping small particles in the AlOH floc that forms. pH depressions to 6.5 from repiration may cause aditional floc formation, while pH rise will resolubilize the settled floc, thus you may see additional formation of settled solids after you remove what is there now. Filtering through a very small pore size filter might work, but the filter will quickly blind off unless you have a deep bed filter like used on a swimming pool.
Good luck
