AFAICR (I'm not at my Tiny Core box), these are the steps I went through to install Tiny Core on a USB stick. I had already determined that I could boot from CD and connect to the Tiny Core repository via the appbrowser on the box I was using:
1) Downloaded Tiny Core ISO and burned it to CD (did this on a Windows box).
2) Booted the machine I use for Tiny Core off the CD. USB stick was already be plugged in. Used bootcodes "waitusb=5 base norestore" [no quotes].
3) Mounted the CD (usually it unmounts after boot) using the mount tool. This gives a source for the files you want to install.
4) Opened a terminal and typed sudo su (gets root terminal).
5) Typed "usbinstall" [no quotes].
6) At this point the installer starts asking what you want to do. I chose the ext2 format and to keep a log.
7) If you don't have the Tiny Core extensions you need to complete the install, the installer will tell you at this point and exit. Download the extensions it tells you with the appbrowser (use 'install') and run usbinstall again, repeating the above.
Follow the subsequent installer directions. You have to tell the installer where your Tiny Core files are (on the mounted CD, I forget the directory name) and where the USB stick is (I think sda).
9) The installer should create a boot directory (with Tiny Core files), a directory for logging, and the tce directory (with optional directory underneath) on the USB stick.
10) When the installation is complete, unmount the CD and remove it, and then reboot. Your machine should boot off the USB stick. Note that you have to have your BIOS setup for this (I had to tweek mine a bit).
11) If the boot goes OK, at that point the appbrowser should download any extensions you want to the tce/optional directory on the USB stick, and find them there on subsequent boots.