Depth is just showing the 3Dness of something, showing how it recedes into the canvas/screen instead of being flat. It sounds like you're not talking about "using depth", but rather showing depth through colour. Depth is a consequence of using colour, overlap, and shadows, etc (or some subset of those, e.g. just colour).
The general gist of using different colours to show depth is to assign different colours, typically fading from a bright "close" colour to a duller "far away" colour, to different levels of depth. This means you need to understand which parts of your character are closer and farther away. So, if the left arm is closest in one sprite, it gets the "close" colour(s), and if it's farther away in another sprite, it gets the "far away" colour(s). If two things are at the same depth (i.e. the same distance from the viewer), then they have the same colour(s).
