Begin by logging into roundcube.
At the bottom of the screen, click the "Select All" icon.
At the top of the screen click the "Move message to trash" icon
This may take several minutes, depending on the number of messages in the Inbox.
If you have not previously set roundcube to "Clear Trash on logout" (under Settings > Server Settings) you will need navigate to the Trash folder to finish deleting the emails.
Click on the "Trash" folder on the left side of the screen.
Click the Select all icon at the bottom of the screen
Click the "Delete" icon to completely delete the messages (note there is no warning that messages will be deleted).
This process may take several minutes too, depending on the number of messages being deleted.