Have you tried:
■flushing any caching plugins you might be running, as well as server and/or browser caches.
■deactivating all plugins (yes, all) to see if this resolves the problem. If this works, re-activate the plugins one by one until you find the problematic plugin(s). If you can't get into your admin dashboard, try resetting the plugins folder by FTP or PhpMyAdmin (read "How to deactivate all plugins when you can't log in to wp-admin" if you need help). Sometimes, an apparently inactive plugin can still cause problems. Also remember to deactivate any plugins in the mu-plugins folder. The easiest way is to rename that folder to mu-plugins-old
■switching to the Twenty Twelve theme to rule out any theme-specific problems. If you can't log in to change themes, you can remove the theme folders via FTP so the only one is twentytwelve. That will force your site to use it.
■manually upgrading. When all else fails, download a fresh copy of the latest.zip file of 3.6 (top right on this page) to your computer, and use that to copy up. You may need to delete the wp-admin and wp-includes folders on your server. Read the Manual Update directions first.