Most hair dyes these days are non-toxic and safe to use while breastfeeding
The results of genetic toxicity, carcinogenicity and reproductive toxicity studies suggest that modern hair dyes and their ingredients pose no genotoxic, carcinogenic or reproductive risk. This is based on an article published in toxicol appl pharmacol. 2010 mar 1;243(2):239-59. Epub 2009 dec 21.