More and more companies are using chatbots in customer service. The large number of chatbots and their interactions with customers produce a huge amount of data, which is useful to track the usage and performance of the chatbot. However, many established performance metrics (e.g., intent scores, conversations per day) could be considered too intuitive to be helpful and are either at a very high level or at the level of single question-answer pairs. Our research aims to address this challenge by presenting a novel approach and system for conversation analysis of customer service chatbots. More specifically, we extend established metrics and concepts with ideas from process mining since every conversation with customer service chatbots can be interpreted as a sequence of discrete steps. This paper presents the methodological foundations for our approach, which we call conversation mining, and demonstrates its potential with first insights into our prototype. Ultimately, we aim to draw the attention of chatbot researchers and practitioners to the value of conversation data by describing a novel approach for automatically processing and analyzing at a process level.