Preheat the oven to 350 degrees F (175 degrees C). Grease two 8-inch round cake pans.

Combine cake mix, eggs, water, and oil together in a large bowl. Beat using an electric mixer on medium speed for 2 minutes.

Split the batter into 6 different bowls and add a different food coloring to each bowl. Pour about 1 cup of each of the batters into the first pan in this order: red, orange, yellow, green, blue, and purple, right on top of one another in the center of the pan. Pour about 1 cup of each batter in the same manner into the second prepared pan in reverse order: purple, blue, green, yellow, orange, and red; you do not need to swirl the colors around at all.

Bake in the preheated oven until a toothpick inserted into the center of the cakes comes out clean, 34 to 38 minutes.

While cakes bake, beat cream cheese, butter, lemon juice, and vanilla extract for filling together using an electric mixer. Beat in powdered sugar until combined.

Beat heavy cream for frosting using an electric mixer until almost stiff; add powdered sugar and vanilla extract. Beat until cream holds peaks.

Remove cakes from the oven and let cool completely in a refrigerator, about 30 minutes.

Spread filling on the top of the bottom cake layer; top with the top layer of cake. Frost and decorate as desired. Keep cake refrigerated.