If possible, stick to your distro's package management facilities such as apt, yum, portage.
Otherwise, make sure that the installed binaries are executable by your target audience:
chmod 755 /opt/bin/chrome # this will do the trick.
adjust the path of the binary as needed.
Also I think you should consult the documentation of the installer because setting correct permissions is one of its responsibilities. I'm willing to bet Google didn't screw this one up (but you may never know).
The easiest way to upgrade your chrome is to download and extract the updater archive to a temporary location and then to copy the files over to the installation folder. The installation folder will vary depending on where the user installed the browser manually or where the distro's package put it. On Gentoo, Chrome lives in /opt/google/chrome. I sucessufully upgraded my copy of chrome from version 21 to 25 using the following commands:
rsync -rEp chrome-linux/ /opt/google/chrome/
This method assumes that a working copy of chrome is installed and copies all files in the update folder while preserving the permissions executability in the original folder. I suggest you use your package manager to re-install chrome before doing the above as root.
Here is what I see in the about dialog, prior to and after the above steps: