Put steam into offline mode and try again.
If you can't put steam into offline mode due to poor/crappy/non-existent internet connection, then close steam and create a file named steam.cfg in your root steam folder. Add the following ...
BootStrapperInhibitAll=enableForceOfflineMode=enable
Steam should now start in offline mode.
If you still can't get to the standard steam gui (likely due to it wanting to update the steam client), then start it up and let it stall, then go to your Skyrim root folder and start TESV.exe directly. It will happily run as long as there is a steam thread running.
If your install has become corrupted and you need to verify it, then it's time to learn from the experience. Zip your steam (or Skyrim) folder up for backup purposes so you can recover it if required without putting steam online.
This bug ridden POS game that Bethesda has lumped us with is too much fun to allow steam's idiosyncrasies to stop us playing.