Lets be very clear. The open source Chromium project is not Google Chrome. They're similar, but not the same.
Chromium does not bundle Flash Player. Google Chrome does.
You can install the standalone PPAPI Flash Player for Chromium, but there's a bug in Chromium that prevents it from detecting the system PPAPI Flash Player (it's probably resolved in daily builds at this point), so you have to launch Chromium from the commandline using the right commandline parameters. If you're just trying to browse the web and not explore experimental browser features, you're probably better off with the generally available Chrome release.
There's a guide to the available Chromium commandline parameters, here:
http://peter.sh/experiments/chromium-command-line-switches/