How can I clear the screen in the MySQL console?

Try your terminal application's screen clear. There is no such thing as a command to clean the MySQL shell.

On a Mac, you can go to menu Edit → Clear Screen or use the keyboard shortcut ⌥ + Cmd + L.

Ctrl+L

will do it - it's a shell command, but it works well in the MySQL console.

Another possibility is to issue a system command.

For Linux:

mysql> system clear;

For Windows (I'd guess. For Linux I'm sure though):

mysql> system cls;