Thyroid issues was the first thing that popped into my head as well. It's not uncommon for thyroid levels to get out of whack after giving birth.
Definitely see your doctor. I'll second Theresa that this is not related to breastfeeding, but it could be post-partum hypothyroidism, among other things.
Breastfeeding does not cause these things.
You need to go to the doctor and have some tests done - could be a number of things.
Sometimes hair falls out soon after the baby is born. But this is more because during pregnancy, it has not been falling out, and there is a bit extra hair loss afterwards to compensate. But not what you describe.