How can I communicate with the MCP39F511?
You can interface it to any microcontroller that has a UART. Via UART, the communication protocol can be found in the "Power-Monitoring IC with Calculation and Energy Accumulation" datasheet in the "Documents" section of the MCP39F511 product page.
To learn more about UART, refer to the "Using the USART in Asynchronous Mode" document which can be found under the file name Getting started - USART Asynchronous in the "Product Documents" section of https://www.microchip.com.