You say that you've got a new computer (rather than just upgrading the operating system on your old one) but you haven't indicated how you've tried to transfer the files. Your old e-mails will be stored, on your old computer, in a file called inbox.dbx (If you had your e-mails sorted into various folders, then you'll also have other files such as work.dbx, mother.dbx or whatever other names you used).
You need to search, on your old computer for *.dbx (If that's not too clear on your screen, its star-dot-dbx). Then copy all these files to a CD. Then repeat the search for *.dbx on your new PC. Right-click on one of these files and click 'Open containing folder'. Then insert the CD and copy the old dbx files to this folder. (When you're warned that some file names already exist, click to over-write them. NB: This will delete any new e-mails which you've already received on your new PC! To avoid this, you could try renaming the old inbox.dbx file to something like oldbox.dbx, etc). Outlook Express should now contain all of your old e-mails.
Chris