We never like to say no; we’ll jump through hoops to figure out ways to help people do innovative things in the kitchen. But the short answer here is, no. When you make a white cake from scratch there is often a variation listed with the recipe that allows you to turn it into a chocolate cake by substituting cocoa for a portion of the flour. The key word is substitute.
Generally you decrease the flour by one third and make up the difference with cocoa. But if you throw out a third of your dry cake mix to replace it with cocoa, you’ll also be disposing of one third of the sugar, salt, leavening, and all those yummy preservatives. Alternately, if you added a half-cup or so of cocoa, you would need to increase the volume of eggs and other liquid ingredients to keep the cake from being too dry, and possibly the salt and sugar so that it has the taste you’d expect, and quite possibly the amount of leavening, both because of the increased volume, but also because the cocoa is acidic and will react with the baking soda/powder in the cake mix.
In short, you’d really have to reengineer the recipe, and since you don’t know exactly what is in the cake mix or in what proportions, you really have no basis to work from. The easiest solution is to buy a chocolate cake mix; the best solution is to make the cake from scratch.
Now, having told you that someone would have to reengineer that cake mix (and implying that it certainly wasn't going to be us!), we remembered Anne Byrn, the Cake Mix Doctor, who has single-handedly carved a career out of fixing up cake mixes. If anyone could turn a white cake mix into a chocolate cake, it would have to be her. We looked through Chocolate from the Cake Mix Doctor pretty thoroughly and even the Doctor wants you to make chocolate cakes with chocolate cake mix. But there was one recipe for a Red Velvet Cake that uses a white cake mix, cocoa, and, yes, two bottles of red food color. We decided to omit the food color, add a little extra liquid to compensate, and rename it the Cake Mix Doctor's Chocolate Velvet Cake. It only has three tablespoons of cocoa in it, so it's not going to be the most intensely chocolate cake you've ever tasted, but it's got to be better than sitting around with a box of white cake mix and an unresolved hankering for chocolate cake.