To add a custom chat channel to Twilio Flex, developers need to create a new Studio Flow and associate it with the Flex Chat Service. They also need to implement middleware that receives messages from the chat channel and sends them to Twilio's external service, such as Socket.io. The middleware uses the Flex API to create a new channel and configure a webhook for the onMessageSent event of the chat channel. The solution involves creating two functions: `createNewChannel()` to create a new channel and configure the webhook, and `sendChatMessage()` to send messages to Flex once a new chat message is sent from the browser. To test the solution, developers need to build a webchat using Socket.io and add a new POST endpoint to handle the message coming from Flex. They also need to implement a function to send a message to Flex once a new chat message is sent from the browser.