A true red velvet cake is a buttermilk based light chocolate cake. If you want to keep the chocolate flavor, find a recipe that uses cocoa powder and baking soda. The baking soda will give the cooked cocoa a reddish color.
I second the idea of starting with a box mix of strawberry or raspberry cake. Instead of using a liquid color, try using a gel color instead. The color is much more concentrated and a little goes a long way without changing the flavor too much.
Bear in mind that "true" colors require lots of pigment and so tend to give a funny taste. You may want to compensate for that by adding in a bit of extra flavor if you decided to go with a non-chocolate cake.