You need to locate the WAB (Windows Address Book) file to recover your contacts, it's default location is:
C:\Documents and Settings\USERNAME\Application Data\Microsoft\Address Book\USERNAME.wab
A tool for extracting e-mails and attachments from the DBX files can be found here:
http://macallan.club.fr/index.htmDefault location of the files is:
C:\Documents and Settings\USERNAME\Local Settings\Application Data\Identities\{GUID}\Microsoft\Outlook Express
Your information will not appear in Outlook Express on the other computer until it is properly imported into it.