Some companies have a decent selection of bikes which you can configure and customize with their frames (or a small subset of other people's frames) -- Some of these are Rivendell, Public, Peter White Cycles and similar. The frame and color selection is typically quite limited though (to the frames that the company makes), and generally more expensive than comparable (when applicable) pre-built bicycles.
If you want to have full customizability, you'll generally have to select out the parts (including the frame (custom-designed or off the shelf)) individually and then put them together on your own. If you want custom coloring on a frame, generally, you have to find a shop which can paint it (or powder coat it or something -- note that aluminum and carbon frames are harder to re-color than steel frames). Then, go to your local bike shop and have them put together the parts you selected. There are so many frames and what not out there that I don't think anyone can offer anything but a small selection of bikes within reason as pre-built.
Note that this generally requires some expertise - you need to know what size frame you want and what parts are compatible and will fit you. So, if you choose to do the full custom route, its best to do it with the assistance of a bike shop if you don't know what you're doing. Also, expect to spend at least thousand on this (typically significantly more than this), if you choose to pursue this route (even without a respray or powder coating of a frame).
An alternative is to start with a bike you mostly like, and replace what you don't like. This is far more economical in most cases.