Is it safe to take skin, hair, and nail vitamins (5000 mcg of biotin) with doxycycline (for acne)?

Not recommended. because biotin will decrease the level or effect of doxycycline. Always talk with your doctor if you have any questions regarding a drug's side effects and interactions.
Acne. Your doctor can help you with your acne. This is a better and safer route to take rather than the so it yourself approach.