How does permanent hair dye work?
(Q&A by Heather Peterson)

The process of hair dying is actually just one big chemical reaction. In order for the dye to permanently change your hair color, it must first get through the cuticle and then reach the hair shaft. The dye itself is made up of two components that are separately packaged and then are mixed together before the application. The packages contain hydrogen peroxide and an ammonia solution of dye intermediates and performed dyes which are called couplers.

In order for the hair coloring to stay in the hair shaft, the hair's normal pigments must be removed so that the dye intermediates can replace them. This is what the peroxide does. It bleaches out the hair color and then allows the new pigments to take its place. Once the cuticle is opened and the pigments have been removed, the dye can then take the place of the pigments in the hair shaft which will then change the hair's color.

The intermediates develop a color upon oxidation, but the couplers don't oxidize. The couplers actually react with the oxidized intermediates which help to obtain more color. Once the chemicals are mixed and applied to the hair, the hair begins to swell because of the ammonia and then the scales of the cuticle begin to separate. This allows the dye to seep into the hair shaft after a period of time, thus changing the hair's color.

SOURCE: http://pubs.acs.org/cen/whatstuff/stuff/7811scit4.html