I am running Windows 8 on an HP laptop, I installed the student version of Revit 2013 (which I have successfully installed and used on Windows 7 on the same laptop). When I open Revit, it goes to the splash screen, briefly shows the open file page, and then it crashes.
I have tried deleting the license in the FLEXnet folder of C:\ProgramData so I could use the transfer license utility in case it was a license issue but I can't even get that to open. I've also repaired the installation of Revit to no avail.
I am running the same version of Revit 2013 on Windows 8 on a dual-booted Mac Mini and I was able to install update 2. So far it seems to be funcitoning normally that way.
The program is not supported on Windows 8. But what you can try is to make sure you have full access to the file and folder structure. You should have full admin rights and disable UAC. Run the software as administrator and in Windows 7 compatibility mode.