I think you'll find the Core Book, downloadable from here http://tinycorelinux.net/book.html
Chapter 2 shows how to install Tinycore, or core for that matter..
In it's simplest form, core is only two files available from here http://tinycorelinux.net/7.x/x86/release/distribution_files/
Copy both files to a convenient and suitable location. Modify your bootloader config file to suit, apply any necessary boot codes (see Chapter 10).. then boot to core.
I could be wrong, but afaik tc-install is a gui application available after booting to tinycore (or tinycoreplus). Booting to tinycore and running tc-install will allow you to install core on almost any media (that you can write to).
However " tc-install.sh " is available from the command line, but if you're at this stage in the game I wonder why you cant just install the bootloader and core manually..
After core and a suitable extension directory ( /tce )is installed, you might find " tce-ab " very useful to install packages (extensions) from the command-line.
All this information is available and explained in much better detail in the Core Book mentioned earlier.