Have you already made a Steam account and tied Skyrim to it?
Try this:
Do not install Steam from the Skyrim disk.
Instead, download and install Steam from the official site. Launch it and log on with your account.
Wait for the platform to completely update itself. It shouldnt take a while.
Once it is done, leave it running then launch the Skyrim disk and install.
After you install your game, verify your game cache files. (https://support.steampowered.com/kb_article.php?ref=2037-QEUH-3335)
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Do you have any of those two folders in your HDD ???
C:\Program Files\Steam\steamapps\common\skyrim\data ----------------------- Win 7 32 bit
C:\Program Files (x86)\Steam\steamapps\common\skyrim\data ----------------------- Win 7 64 bit
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Sometimes Steam will appear to be functioning correctly, except certain games will be missing from your list.
If you are absolutely sure that you are logged in to the correct Steam account, then the issue is most likely that your filter is set incorrectly.
Please log in to your Steam account using the Steam client and ensure that all games are visible by clicking View > All Games on the upper-left corner of the Library > Games list.
Another possibility is that some ports are being blocked by your firewall and/or router. Since Steam cannot connect on all of the ports it uses, it may run into a problem verifying one of your games.
To see the list of ports and to make sure that all of the ports for Steam are open.
Required Ports for Steam: https://support.steampowered.com/kb_article.php?ref=8571-GLVN-8711
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I just purchased a game but I don't see it in the Library list. Where is it ???
You may have to log out of your Steam account and log back in to Steam through the Steam client in order for the game to appear in the Library list.