A married man wants a ta'aruf (getting-to-know) with me with a view to polygyny. What should I ask?

Ask about the things that determine fairness and stability: his means, the existing wife's knowledge, and how it will be put into practice.
Important questions include: Has the first wife been told? Will he apply for the Syariah Court's permission, and when? What are his current income and dependants? How will the rotation of nights, separate accommodation and maintenance be arranged? Why does he want to take another wife? Involve your wali in the conversation and ask to meet his family. Avoid vague promises or suggestions to marry abroad without permission, as that can be an offence and harm your rights. Allah makes fairness a condition; if you fear you will not be fair, then one is enough (an-Nisa' 4:3). See conditions and prospective second wife.