1. Remove currently installed versions of Flash.
sudo apt-get purge flashplugin-nonfree gnash gnash-common mozilla-plugin-gnash nspluginwrapper swfdec-mozilla
Visit http://labs.adobe.com/downloads/flashplayer10.html to update the filename given under if necessary.
2. Download the 64-bit Flash plugin and extract it:
wget http://download.macromedia.com/pub/labs/flashplayer10/flashplayer_square_p1_64bit_linux_091510.tar.gz && tar xvfz flashplayer_square_p1_64bit_linux_091510.tar.gz
3. Move the file in /usr/lib/mozilla/plugins for a site-wide install (or in ~/.mozilla/plugins for a single-user install):
sudo mv libflashplayer.so /usr/lib/mozilla/plugins/libflashplayer.so
At this stage, you should restart Firefox and verify that the plugin is installed properly by typing about:plugins in the URL bar.
http://dailystruggles.org/installing-flash-plugin-for-64-bit-system-in-karmic-koala/
[...] Installing Flash Plugin for 64-bit system in Karmic Koala [...]