Normally I would install the Windows Steam in PlayOnLinux, then run Steam and click install for Skyrim.
When its done downloading normally Steam will install the necessary libraries like vcrun2008, d3dx9 etc.
Is that were you are stuck? I have a feeling its failing to launch because the libraries are missing...
I installed the following PlayOnLinux Packages (Components, LIbraries):
- POL_Install_corefonts
- POL_Install_d3dx10
- POL_Install_d3dx9
- POL_Install_tahoma
- POL_Install_vcrun2008
- POL_Install_xact