The main ingredients involved in any coloring process are hydrogen peroxide and ammonia. The other ingredients are dye precursors (and moisturizers to help protect the hair from damage). The ammonia solution makes the hair cuticle (the outer layer) swell. This allows the other chemicals to penetrate into the cortex (core) where the natural melanin pigments are stored. Hydrogen peroxide (the developer or oxidizing agent) helps initiate the coloring process. Depending on its strength it bleaches the hair pigment and it oxidatively couples the dye precursors (aromatic amines) into larger dye molecules that can not escape the cortex. This makes the hair color permanent. So when you go from blond to a darker color - say brown- permanent hair colors use the interaction between the ammonia and the peroxide to create a new color base in your hair shafts. If you go the opposite direction - from brown to blonde- first bleach is used to strip the color from the hair. Then the ammonia-peroxide solution creates the new color and deposits it in the hair shaft.

If you use a semi-permanent color the hair is only coated with color rather than deposited into the hair shaft. Some of the hair dying products for men have lead (II) acetate in it. It is used to darken hair by reacting with the sulfur present in the hair color product and in the amino acids cysteine and methionine.The product of the reaction is black lead (II) sulfide. Now, if you want to go into more detail with the lead sulfide there is a web site that discusses the quantitative analysis of lead. Go to lead
Maybe this can give you an idea.