There are a few main connections between your thyroid and acne. Namely:
Low thyroid function increases estrogen dominance (both via lower progesterone and lower Sex Hormone Binding Globulin)
Estrogen dominance in turn increases hypothyroidism
Low thyroid decreases Vitamin A production
For more on this, check out the blog post Acne and Your Thyroid: http://www.drlaurendeville.com/articles/acne-thyroid/