Is it safe to take a hair, skin, and nails (contains biotin and vitamin c) pill, while I am taking doxycycline 100mg for my acne?

Should be OK... Fortunately Doxycycline does not have too many drug interactions and should not interfere or be significantly affected by your use of the supplement described in your question. But like all else, unexpected reaction can occur and so if you experience any significant symptoms, consult doc. Good luck.
Safe. But a waste of money. If doxycycline works for you continue. For adult women, menstrual patterns and exact details of any birth control are central. 90% of women have better results with spironolactone, which is also better tolerated and safer. Biotin and Vit C are totally useless waste of money.

Related Questions

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

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. Read more...
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. Read more...