First of all, let's list the possible elements in the supply chain you will manage: logistics (transportation, warehousing, at least); procurement; supply chain management systems (procurement, production, fulfillment and distribution processes, including EDI); CRM; ERP.
Your primary requirement is to manage the flow of goods, information and services in order to deliver maximum value to the customer while minimizing the cost to the company.
It is a cross-functional process: within the company and between the trading partners.
Depending on the company, these requirements may be more or less. Note how closely this list touches on other pieces of the business: ERP, CRM and EDI depend on the your CIO for a platform. Manufacturing is closely involved with ERP. Marketing could be heading the CRM function.
So you really need experience with each of the major areas. Ideally you will have worked in procurement, transportation, warehousing/distribution. Exposure to information technology, manufacturing, EDI, marketing or finance are all a plus.
You will have several direct reports, so management experience is vital. Experience with third-party logistics is important. If you are moving into a new company, then industry experience is needed.
Finally, my list of Must-Have attributes: