Easy steps:

Use the steps in terminal window...........

(1) Remove currently installed versions of Flash.

sudo apt-get purge flashplugin-nonfree flashplugin-installer gnash gnash-common mozilla-plugin-gnash swfdec-mozilla

(2)Download the 64-bit Flash plugin and extract that file:

wget http://download.macromedia.com/pub/labs/flashplayer10/libflashplayer-10.0.45.2.linux-x86_64.so.tar.gz && tar xvfz libflashplayer-10.0.45.2.linux-x86_64.so.tar.gz

(3)Put the file into /usr/lib/flashplugin-installer/libflashplayer64.so or /usr/lib/firefox-addons/plugins/libflashplayer64.so or (or in ~/.mozilla/plugins for a single-user install):

sudo mv libflashplayer.so /usr/lib/firefox-addons/plugins/libflashplayer64.so

(or)

sudo mv libflashplayer.so /usr/lib/flashplugin-installer/libflashplayer64.so

(4)mk into it.

sudo mkdir /usr/lib/flashplugin-installer

(5)Install the link (60 is the priority, you can use any value 0-100)

sudo update-alternatives –install /usr/lib/xulrunner-addons/plugins/flashplugin-alternative.so xulrunner-addons-flashplugin /usr/lib/flashplugin-installer/libflashplayer64.so 60

Restart the firefox .. now check youtube.

All the best.........