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.

It will be very difficult to match the range of intense colors that food coloring can provide with only fruits and vegetables as coloring sources. Blue will probably be impossible. Pink to reddish-purple may be achievable.

Beet juice would be red-to-purple, and probably intense enough to color moderately without adding too strong a flavor. For a more pure reddish hue, I would suggest making a raspberry cake, but that would tend to be more pinkish than red--getting intense color is difficult.

If you do choose to make a raspberry colored cake, I would recommend getting a recipe that is raspberry all the way, instead of substituting. For beet juice, I would replace some of the water or milk with beet juice, up to about half of its volume, although if you are replacing milk you may loose some tenderness.

There is no natural food in existance to the best of my knowledge—with the possible exception of blue corn—that is blue.

Blueberries are actually a deep purple inside; it is an irridescent effect on the surface that makes them appear blue. Trying to color a cake with them will give you purple. Think of the color of the stains inside a blueberry muffin...

I did SEVEN different experiments to create a pink cake for my daughter. Beet juice will lose it's color when baked. Fruit purees will keep their color, but they aren't great for the texture of the cake. Freeze dried fruits, ground in to a powder, will color the cake, but it will be faint. Hopefully this will save anyone who reads this a few wasted ingredients. If you want to see pictures of my mad experiments... http://www.luckynumber3.com/2016/02/pink-cakesixth-seventh-attempts.html.