I just bought Dragonborn and Skyrim updated. I hadn't played it in a while, so I don't know if the update caused the problem. Anyway, here's what happens:

Upon clicking "Play" (or using the .exe in the folder), a taskbar tab for the launcher opens. The window itself never appears, and all Steam operations immediately stop responding. No error message appears. Left alone long enough, the program closes itself.

I've tried:

• Unsubscribing from all mods
  
• Verifying integrity of game cache
  
• Running secondary installers (DotNetFX, DirectX10, VCRedist)
  
• Using RegEdit to check Steam permissions
  
• Temporarily disabling antivirus (Avast!)
  
• Deleting .ini files in C:/User/My Documents/My Games/Skyrim
  
• Updating drivers (Already up to date)
  
• Updating Steam (Already up to date)
  
• Reinstalling game

I'm running as Admin.

OS: Windows XP Professional SP3 (32 bit)
CPU: AMD Athlon 64 X2 5200+ (2.7 GHz)
GPU: Radeon HD 6870 (1GB GDDR5)
RAM: 2GB GDDR2

Try downloading SKSE, there's a really good install guide for it on youtube and run it through the SKSE starter and not the steam starter. Might work.