I have a Windows 2000 machine.  I want to customize my logon.scr
(screensaver) file to show my companies logo instead of the generic
windows logo when the screensaver initializes.  I do not want to
install any 3rd party software, just somehow update the picture that
this .scr files uses.
What is the easiest way to do so?

Microsoft has buried the "Starting Up...." logo/splash screen in
C:\WinNT\System32\Ntoskrnl.exe file to make it fairly inaccessible.

The easy and safe way to change it is to use third party software
(BootSkin, wincustomize.com, www.startupscreen.net, etc.) But since
you want to do it yourself read the steps at the bottom of this
answer.

Otherwise, you can safely and easily change your Windows boot screen
with BootSkin. It's free. It allows you to replace your Windows boot
up screen with ones you can download from websites such as
WinCustomize.com or create yourself. BootSkin doesn't patch the
Windows 2000/XP kernel. Nor does it require the user to download
replacement Windows 2000/XP kernels to do so.

Get it free at http://www.brothersoft.com/Desktop_Utilities_Start-up_Screens_BootSkin_25715.html
(Note: Yes, there are a lot of ads at the top of this web page - you
have to wait for them to load before you can get BootSkin. Sorry.)

HOW TO DO IT YOURSELF:

---------------------------
 1. Edit your logo
---------------------------
Make your logo 640x480 in 16 colors or less. Then use the automatic
bootscreen bitmap creator at:
http://server1.belchfire.net/Inno_Resize/index.php

---------------------------
 2. Download a tool temporarily
---------------------------
(Read the link at step 3 for more information about this tool.)

You'll need this tool temporarily (you can remove it when done):
http://www.users.on.net/johnson/resourcehacker/

---------------------------
 3. Edit Ntoskrnl.exe
---------------------------
WARNING!! Dangerous step, not recommended.

But, if you want to "do it yourself," edit the NT/2000 system file
Ntoskrnl.exe following these instructions:
http://www.beemerworld.com/tips/winy2ksplash.htm

If you change your mind .. 
www.startupscreen.net
www.wincustomize.com

Wow, thanks for all the info.  But these steps seem to point me in the
direction of changing my startup splash screen, not the screensaver. 
Unless they somehow go hand-in-hand.

To edit the .scr file you're going to have to download some 3rd party
software(for the editing only)then you can uninstall it when you're
finished.  Check this program out:

http://www.freedownloadscenter.com/Programming/Editors/Resource_Tuner_SimpleDownload.html

I've edited my logon.scr file with it. Give it a try, I'd make a
backup of your logon.scr file anyway.

then you can use the registry editor to change the actual .scr(the one
your going to use as your non-logged in user .scr file for instance
logon1.scr, that's what I named mine after editing it.)

http://support.microsoft.com/?kbid=314493

hope this is kinda what you were after.

Regards,