Each organization is an individual case. However, when working with organizations implementing ISO 9001:2015 I follow, as general rules, this framework:
Customer and other interested parties satisfaction objectives;
Product or service objectives;
Process performance objectives.
For each objective I want to know what the actual value is and the trend. I always use graphics, and compare results with the target. Whenever possible I like to use control charts in order to help decide if the system should be changed or not in order to meet targets.
Personally I prefer to link performance and processes, but my experience is that most organizations prefer to link with departments.
About measuring customer satisfaction I see organizations using surveys, using interviews with customers, using experts opinion in magazines, using consumer evaluation in websites.