Currently, there's no way to clear your chat completely, and messages from previous streams will still be present when you launch a new one. Furthermore, there's no way for an Artist to delete previous messages from their chat.
While we're constantly looking into ways to improve your experience, at the current moment this is not something we implemented yet. As a workaround, you can fill up the chat with more recent messages in order to hide the messages you don't want others to see.