There has been great recent advancement in human-computer chat. However, proper evaluation currently requires human judgements that produce notoriously high-variance metrics due to their inherent subjectivity. Furthermore, there is little standardization in the methods and labels used for evaluation, with an overall lack of work to compare and assess the validity of various evaluation approaches. As a consequence, existing evaluation results likely leave an incomplete picture of the strengths and weaknesses of open-domain chatbots. We aim towards a dimensional evaluation of human-computer chat that can reliably measure several distinct aspects of chat quality. To this end, we present our novel human evaluation method that quantifies the rate of several quality-related chatbot behaviors. Our results demonstrate our method to be more suitable for dimensional chat evaluation than alternative likert-style or comparative methods. We then use our validated method and existing methods to evaluate four open-domain chat models from the recent literature.