It's worth noting that Outlook Express is not an independent program, but a part of Windows, and that Windows XP is the last version to come with this particular email client. Therefore, how you move your inbox depends on what version of Windows you're moving it to.
But first, you have to copy the messages off of your old PC. I'll assume that PC uses XP; if you're using an older version of Windows, these directions won't be an exact fit, but they'll be close.