What can I add to a cake mix to make it a bit more dense with a finer crumb? I want to stack a few layers with a fair amount of filling and I want it to be able to support the weight but still be moist.
First of all, your cake shouldn't support the upper tiers. The dowels in the tiers support the upper tiers. I use cake mixes all the time and generally don't doctor them up except for Red Velvet, in which case I make my own buttermilk using milk and vinegar and use that for the liquid instead of water. Also, for white (vanilla) I use the whole eggs instead of just the whites. That gives it a much better flavor. I always get rave reviews on my cakes using box mixes and the other flavors I use are Spice, Chocolate, Lemon, Confetti and recently a vanilla cake champagne as the liquid instead of water.
As stated above, you never want to stack more than 6" of cake at most without dividing it into tiers and putting in structural supports.
I add an extra egg per cake mix (4 instead of 3). I also recommend either dropping the filled cake pan on the counter a few times or whacking the bottom of the pan instead, to get the largest air bubbles to rise to the surface and pop, so you won't have big air pockets in your cakes.
If necessary, you can have a "hidden" tier, where the cake on top is the same size and you stack them before icing and decorating, but you have support for the upper layers. I did this on this wine cask cake: