It really depends on what kind of dye.
Most of the dye I use doesn’t actually strip pigment from the hair, so it’s not terrible for it.
But bleach kills my hair. The only reason it ever looks healthy is because when it gets too damaged I just cut that part off, and also condition it heavily to heal from the chemical.
I recently bleached my hair twice in two days, and that screwed it all up.
This might explain it a little better than I did:
The only thing that could kill your hair is bleach… when you dye it, no matter what color red, pink, blue, black, you’re adding pigment to it. It’s stripping pigment aka bleaching ruins the natural quality.