Sheet metal roofing fastened to the underside of deck joists makes a great, weatherproof ceiling for applications like yours. You can order it in whatever lengths you need to extend from the house wall out to a few inches beyond the edge of the deck. Fasten the metal with special roofing screws that have neoprene washers underneath the heads, with the factory-painted surface of the roofing facing down. For best results you’ll need to install this metal on tapered strips of wood that ensure a consistent slope away from the house for good drainage. You can direct the water to run towards the front or sides of the deck, whichever is more convenient. Either way, overlap the seams one notch further than you would if the metal was installed in the usual way, just to make sure there are no leaks. You might also consider installing eavestrough around the perimeter to stop water from splashing onto things stored below. You should also consider blocking the area around the perimeter of the deck ceiling to keep out nesting birds and other animals.