Preheat oven to 350 degrees F (175 degrees C). Grease a 9-inch square cake pan.
Mix chocolate cake mix, water, vegetable oil, egg, and instant coffee together in a large bowl until well blended. Pour into the prepared cake pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 25 to 35 minutes.
Whisk milk, evaporated milk, chocolate pudding mix, and Irish cream liqueur together in a bowl until smooth. Place in the refrigerator.
Cool cake in the pan for 5 minutes. Invert onto a wire rack to cool completely, about 20 minutes. Cut cooled cake into 1/2-inch cubes.
Spread half of the cake cubes in the bottom of a trifle bowl; brush with Irish whiskey. Layer half of the pudding and whipped topping on top. Repeat layers with remaining cake cubes, Irish whiskey, pudding, and whipped topping. Dust trifle with cocoa powder.