Nevin, either aluminum or fiberglass window screen works fine. A couple of layers is plenty.
As far as how much gravel, the pots that you use will control this somewhat. If you can find clay pots that will fit into the box with a couple of inches of gravel, this is perfect. You can put a little more gravel in to keep the pots in place, if you wish. The pots should be the kind with holes in the bottom and, like the flower box, they need a good drainage system also. You can use a little leftover screen for the bottom of the pots and add gravel there also.
My daughter has a box that I built about 10 years ago from cedar. She has it outside her kitchen window and she waters with the sink sprayer. The paint is a little faded, particularly the stencilling, and we take the box into the barn for the winter. There is a cedar 2x4 cut on a 45 degree angle longways and the piece with the long edge to the outside and the open space pointed up that is screwed to the house and the piece with the long edge to the outside that's screwed to the box with the space pointed down fit together. Hope you understand that but it's a lot easier to draw than describe in words.