Preheat oven to 325 degrees F (165 degrees C). Grease and flour three 8-inch cake pans.

Beat cake mix, milk, oil, eggs, and vanilla pudding mix together in a large bowl with an electric mixer on low speed until just blended. Scrape the bowl, increase mixer speed to medium, and beat until batter is smooth, about 4 minutes. Pour batter into prepared cake pans.

Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, 20 to 25 minutes. Cool in the pans for 10 minutes before transferring cakes to a wire rack to cool completely.

Beat cream cheese, confectioners' sugar, and white sugar together in a bowl with an electric mixer until smooth; fold in whipped topping until frosting is smooth.

Place 1 cake on a serving platter. Spread frosting on top of the cake; top frosting with a layer of strawberries. Place 1 cake atop strawberries. Frost cake and top with a layer of strawberries. Place remaining cake atop strawberries, frost the top and sides of the cake, and top with remaining strawberries.