All of the messages in Outlook Express are kept in one directory on your harddrive. If you do a search for .dbx (right click on Start, select Search... from the popup menu, enter .dbx in "All or part of the file name", and click on the Search button), you should be able to find that directory. Each folder in Outlook Express is shown as a dbx file.
If you find two or more different directories, the one with the more current files is the one to use.
Copy the files to the new computer and then use File - Import - Messages to import the messages into Outlook Express.
If it is an empty Outlook Express, you can just copy the files to the appropriate directory for that Outlook Express instead of importing them and they will show up in Outlook Express when you start it.