Preheat the oven to 350°F. Lightly grease an 8 x 8-inch baking pan.
In a large mixing bowl, stir together the sugar, flour, baking powder, salt and cocoa. Add the milk and vanilla; beat till smooth. Pour the batter into a lightly greased 8 x 8-inch pan.
In a small bowl, mix together the white and brown sugars and cocoa. Sprinkle evenly over the cake batter in the pan. Pour the coffee over all. This may look kind of strange and messy, but never fear; everything will turn out fine.
Bake the cake for 40 minutes, or until the top springs back when pressed gently. Remove from the oven and cool on a wire rack. This cake is best enjoyed warm, between half an hour and an hour after you've taken it out of the oven.