When you run the installer, an "Installer Log" text file should appear on the desktop. Make sure to upload that here.
Also, check to see if the installer is prompting you for Administrator Permission (this is often a pop-up window). Sometimes programs will ask for Administrator Permission and if you click elsewhere, the window won't automatically pop-up. Check your taskbar at the bottom for any popup windows (it should look like a 'shield'). If this doesn't work, remove the installer and any X-Plane contents by dragging to the Recycle bin, and try re-downloading the installer from X-Plane.com and start again.