Colored cake without food coloring?
As a note I'm answering this question based on avoiding artificial food dyes (Red #40, Yellow #5, etc) rather than all food coloring.


Raspberry puree will produce a pink result, and I have sucessfully done a very dark pink with ground freeze dried raspberries. Achieving the depth of color needed to provide a true red or blue would require not only several food sources to blend the color correctly but a large quantity of, for example reducing juice to make it more concentrated since what you are going for is a color concentrate. 

Another option is to let someone else do the blending and extracting for you. Chocolate Craft Colors and a few other companies make plant derived food colorings, and I like the chocolate craft ones since they disclose all of the plant sources (cabbage in the blue, and beets with carrots for the red) and they sell liquid, paste, and powder so you can achieve the color you want without throwing off the balance of the recipe.