Originally, you had to first defeat normal Baal in Softcore in order to create Hardcore characters on your account. That method still works, but is not necessarily required. There are now alternate ways of enabling HC that are completely legitimate to use.
[1] One way of enabling hardcore is logging onto an account that can already create HC characters. Asking a forum member to create a "dummy" account for you to log in to is one way of doing this.
[2] If you don't have access to such an account the easiest way is to go through the following steps. I've tried to make this as simple as possible due to some people having inhibitions against messing with their registry.
1) Go to the Start menu, and open the program called Notepad (it's in the Accessories folder of the start menu).
2) Copy and paste the text below into a newly created document:
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Windows Registry Editor Version 5.00
[HKEY_CURRENT_USER\Software\Blizzard Entertainment\Diablo II]
"AllowHardcore"=dword:00000001
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3) Press Ctrl+S (or alternatively File
Save As…) and save the document with the name allowhc.reg making sure the Save as type: is "All Files".
4) Double click on the file you just created, click YES that you do want to add the information to the registry, and click OK afterwards.
5) Proceed to create HC characters at your leisure.
Taadaa.