You're using the ESF shortcut? It places one on the desktop when you install I believe.
Right click on the ESF shortcut, and click properties.
Click on the Shortcut tab, and in the target box, add to the end of it -console.
This enables the console in-game by pressing the ~ key.
Yours should look something like mine now, which is
C:\SIERRA\Half-Life\hl.exe -console -game esf
(Notice the path may be different).
If that doesn't work, there's a different way to do it, but it's more complicated and long, and this should work.