As the title says, Flash Player is no longer natively supported in Chromium web browser, if you want to use Chrome (Flash Player in Chrome), you will have to download the Google version:
Go here and select the Ubuntu version you require, either 32 or 64 Bit:
See also (old but still actual):
I recommend you install this package too but I am not sure if it is really necessary although it doesn't hurt as I installed it when I installed Chromium!:
sudo apt-get install pepperflashplugin-nonfree
Another thing (this is old too btw), take a look here for DRM protected playback:
In Chromium, some Flash vids would play ok(ish) while others wouldn’t, after I installed GC, I didn’t (and still don’t) have any proiblems with Google Chrome playing Flash vids, it is better than FireFox imo!.