I bought two beets, wrapped them in foil, baked them in the oven for 90 minutes, cut them into thin slices, and pureed them in a blender. I used this recipe (the recipe is for a layer cake, but I used it for cupcakes instead):
3 1/4 c. cake flour (or all-purpose)
3/4 c. cocoa powder
1 1/4 tsp baking soda
1 1/2 tsp salt
1 c. canola oil
2 1/4 c. sugar
1 1/3 c. pureed roasted beets (about 2 large beets, or 3 small ones)
3 large eggs
1-6 tablespoons red food coloring
1 T. water
1 1/2 tsp vanilla
1 1/4 c. buttermilk
1 tsp vinegar
1/4 cup melted dark chocolate (I omitted this)
Preheat oven to 350
Cooking time for cake: 30-35 minutes
Cooking time for cupcakes: 20-25 minutes
If your making a cake, grease three 9" round cake pans, line with parchment, grease the parchment and then dust with cocoa powder. If your making cupcakes put cupcake liners in your pan.
Sift together flour, cocoa powder, baking soda and salt. Set aside. Mix together canola oil, sugar and beets on low speed. Add eggs one at a time, and beat until well mixed. Add red food coloring and vanilla to oil mixture. In a separate bowl add the vinegar to the buttermilk. Then alternately add the dry ingredients and buttermilk, beginning and ending with the dry ingredients. Finally mix in the melted dark chocolate. Bake until the cake springs back when you lightly touch it (or when toothpick inserted into center comes out clean). Cool ten minutes, then remove from pan and continue cooling on a wire rack. Frost.
Overall rating out of 5:

Terrible. This is not red velvet cake. This is chocolate beet cake, which I guess is fine if that's what you want. Even though I didn't add in all that melted dark chocolate, the chocolate was strong and very bitter (far too bitter). The consistency was oddly chewy. They didn't even smell good when I took them out of the oven. The color was more of a brown than a red, which I expected. I didn't even bother to frost them.
I definitely want to try this again, but with a different recipe next time.