Acid dyes are made to work with fur & feathers. 2 lines are available at
Jacquard's and Dharma Trading house brand, both are good.
1/8th to 1/4 teaspoon is enough to do a cape or a couple pieces of deer fur. The acid is from white vinegar. Google Acid dye and you will find other companies, Veniards has a line geared to fly tying, but their product is very inconsistent .
Here is a video of Davie McPhail demonstrating how to dye feathers.