Your general approach should work, but I'd highlight some details.
Instead of your given list of colors, generate a number of color "bins" in the color spectrum to count pixels. Here's another question that has some algorithms for that: Generating spectrum color palettes Make the number of bins configurable, so you can experiment to get the results you want.
Next, for each pixel you're considering, you need to find the "nearest" color bin to increment. You'll need to define "nearest"; see this article on "color difference": http://en.wikipedia.org/wiki/Color_difference
For performance, you don't need to look at every pixel. Since image elements usually cover large areas (e.g., the sky, grass, etc.), you can get the result you want by only sampling a few pixels. I'd guess that you could get good results sampling every 10th pixel, or even every 100th. You can experiment with that factor as well.
[Editor's note: The paragraph below was edited to accommodate Mike Fairhurst's comment.]
Averaging pixels can also be done, as in this demo:jsfiddle.net/MUsT8/
