Sometimes Customer Marketing falls under Product Marketing, sometimes it doesn't. If it does, where do you recommend the team focus? If it's a separate team, any tips on how to best work with them?

Product Marketing and Customer Marketing go together like peanut butter and jelly! I just love to see the two teams working side by side, regardless of who they report into. 

Product Marketing should be the expert on the "target market" - by this I mean the core markets, customer segments and personas the company is focused on. In order to be effective at this, PMM will need to spend time looking inwards (at existing customers and users) and outwards (at prospects, competitors and new markets). 

I view Customer Marketing as the deep dive into that inward focus component. A strategic Customer Marketing team (or Lifecycle team, as it's sometimes called), should be the expert on the customer. Their focus is on deeply understanding the customer, driving product adoption, and influencing lifecycle behaviours. They can then surface their insights and learnings to the PMM team to use as they identify how to go to market to find more of their best customers.

Additionally, Customer Marketing often acts as the conduit to the customer base for things like product launch. As the owners of customer channels (think email, community, etc), Customer Marketing will play a huge role in the PMM team's launch strategy.