When I use Iceweasel (Debian version of Firefox), I can see Chinese characters found on this page, but not when using Chromium. I instead see rectangles. This is on Debian 7 (Wheezy).
If I remember correctly Firefox ships with some fonts of its own. For Chromium and possibly other programs, you need to install Chinese fonts. I usually use arphic fonts: fonts-arphic-ukai, fonts-arphic-uming