[ Moderator note: Please use the URLs to the codex instead of shortening them like that. ]
If you can't get into your admin dashboard, try resetting the plugins folder by FTP or PhpMyAdmin (read http://codex.wordpress.org/FAQ_Troubleshooting#How_to_deactivate_all_plugins_when_not_able_to_access_the_administrative_menus.3F if you need help).
Switch to the Twenty Eleven 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 twentyeleven. That will force your site to use it.
Manual upgrade. When all else fails, download a fresh copy of the latest.zip file of 3.5 (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: http://codex.wordpress.org/Updating_WordPress#Manual_Update
Also check the Master List to see if you're experiencing a known issue: