What type of skillsets and experiences do I need to build in order to strengthen my career and move from being a Sr. PMM to Director level and above? What type of leadership career tracks do you see people continue their careers?

Tough question since titles vary from company to company. My answer for this question is strictly from a Salesforce perspective. 

My expectation for a Sr. PMM to Director would be that they are able to build a team across all functions of PMM and align cross functional partners (PMs, AR, PR, Campaigns, Content) around an initiative that has a significant business impact. This includes, but not limited to, launching a product, repositioning a product in the marketplace, coordinating a worldwide enablement effort, or driving new, innovative marketing programs across a large area. Each of these requires mastery of your specific area of PMM and the ability to influence senior stakeholders across the business.

There are a couple of ways to think about advancement in a PMM career. You can specialize in a specific aspect of product marketing like inbound or product launches, for example. Or, you can broaden your experiences and amass experience in a variety of areas of PMM. In the current job market, there are plenty of opportunities for to follow either path. My recommendation is to find the aspects of PMM that you truly enjoy and balance that with your interest in learning and practicing different aspects of the role.

In addition to the above -- I would say that "director" job differs from company to company. A director job at a startup is different from a Director job say at a bigger company with larger P&L responsibilities. This also depends on product marketing for different segments because even that varies from company to company. 

In Cisco, a sr PMM for example presents on stages, customer and partner briefings. But I have seen Sr PMM at other companies even within the same industry be completely "behind the scenes" from customers, relying heavily on the product management for message testing etc. 

I would say - identify your segment, that you want to establish yourself, go deep, work on your executive presence, be seen as someone able to contribute to go-to-market strategy, lead a small team, customer facing and sales partnership. 

Off the top of my head, strategic thinking, ability to coach, hire, network, mentor, ability to think bigger than a single product, ability to form executive relationships. There's no silver bullet or training class for these skills - they all only come with time and experience. I see people crash and burn all the time in their first director job. There's no shortcut. It just takes blood, sweat and tears.