Well, if it's just a boxed permanent hair color, you shouldn't get any adverse reaction. Do a strand test, just to make sure. The only thing I can see happening would be that the color might fade even more quickly. I had bleached hair and colored using a purplish/red medium permanent hair color. It looked pretty good for a couple of days and then looked like utter crap. When you bleach, your hair often becomes too porous and won't hold the molecules of color very well. You can combat that by using some filler. Alternately, you can use semi or demi permanent hair color, which holds onto the bleached hair better.
If you're looking to get a more vibrant shade, you could lightly bleach the hair to get 1 to 2 shades lighter, and then color it using the permanent hair color. That is what many stylists do when clients with dark hair want to go much lighter than what normal permanent hair color techniques will allow.