I was thinking about trying to compile Chrome, but I don't have time right now. I think someone has to compile it for TC for it to work (I don't think you can just install from a .deb)
To compile, you'll want to get the extension "tc-compile".
The "google chrome source" web site can provide the source code for download and it'll tell you what libraries/dependencies need to be present at compile-time.
All of the how-to's on making a tinycore extension seem to require more indepth Linux knowlege than I have,* but I've been trying to make sense of them. You can learn more by reading the forums and the wiki.
*specifically, after having compiled a kernel extension I didn't know where the files created by "make" were, and you can't "tar" a file without knowing where it is or what it's name is.