The dye is injected into the skin. It is multiple layers of skin down. Real tattoo ink cannot be dissolved away. It is the permanence of the ink that makes them, you know, permanent. If you could come up with something to dissolve it, the body would do that on its own.
There has been invented a brand of tattoo ink, the name of which escapes me, which breaks down much easier with laser therapy than conventional inks. It initially came in just black color and I’m not sure whether they’ve been able to accomplish anything different in different colors yet, but as far as I know that’s the least permanent ‘permanent’ tattoo option. People should realize that neither laser removal nor natural fading over years (cf. WWII/Korea ear vets’ tattoos nowadays) actually removes the pigment from the body. Instead the pigment is ingested by macrophages in the skin which gradually transport it to the lymph nodes, where it sits indefinitely. Laser therapy breaks up clumps of pigment into more digestible, for macrophages, chunks, and thus speeds the natural fading.
Maybe you you drunk enough paint thinner, get to it from the inside.