Once installed, there's one more step to get Chromium to use the Google Chrome Pepper Flash Player: you need to open /etc/chromium-browser/default as root with a text editor (e.g.: gedit - which I'll use in the command below):
sudo apt-get install gksu #it`s not installed by default in Ubuntu 13.04
gksu gksu gedit /etc/chromium-browser/default
("gksu" is used twice to avoid a bug with Gedit opening a blank file next to our file) 

and in that file, paste the following line at the bottom of the file (after the CHROMIUM_FLAGS="" line):
. /usr/lib/pepflashplugin-installer/pepflashplayer.sh
(yes, there's a dot in the beginning of the line, then a space!)