Coloring braided hair has several effects, not all of them highly desired or positive. Though there are a few things to be considered, removing your braids before coloring your hair is your best bet.
Hair dye will affect the exposed surface of the braids more than the interior of each plait. The coloring will appear solid while your hair is in braids, though once released, the color will be marbled with your previous color.
Permanent color must be neutralized by rinsing and shampooing, otherwise it will continue to process your hair. It is difficult to ensure that all of the color can be removed from the interior of the braid, and the interior strands can be damaged. Chemical burns to the scalp may also occur, particularly with french braid and cornrow styles.
Temporary color is much less damaging than permanent color, as there is no lift or harsh chemical reactions. The longer temporary color sits on the hair, the longer the color effect will last. This can exacerbate the mottled effect if your braids are removed before the color has washed away entirely.
If your braids are created with extensions, the extension hair will be affected by the color. Human extension hair that has been previously colored may not attach the new color in the same way as your natural hair. Synthetic extensions will not be affected by the color at all, though the chemical process may degrade the strands.
Depending on the style of your braids, color can be added by weaving strands of colored hair into the braid, or wrapping the braids with ribbons or thread. Hair color gels such as Manic Panic Dye Hard are made with pigments that sit on top of the hair and wash away easily.