Hello kh335m
Use APPS to Download and install the extension "firefox_getLatest"
then run firefox_getLatest from the desktop menu or from the terminal
After adding firefox.tcz to the onboot.lst (use APPS to accomplish this), reboot.
If Firefox does not load, post the output of running " firefox " from the terminal to find out what the trouble is
If you're attempting to remaster the tinycore ISO then I advise reading the remastering section of tiinycore wiki
Maybe better to explain what it is you're attempting to accomplish when you say "customizing tiny core current"
good luck