Blast, I think you will probably need a holding tank between the downspout and the filter to buffer the water flow.
From what I understand about slow-sand filters, they need a constant flow of water to form and maintain the biological layer ("Schmutzdecke"). Or at least they need to stay full of water once the layer is established. And the layer takes weeks to fully develop.
See the bottom of this page under "Important":
http://www.refugeecamp.org/learnmore/water/slow_sand_filter.htm