That would go something like this if you do not want to disturb the boot loader on the main machine, say if it is a Windows install or you don't have the ability to boot from USB devices. Copy the tinycore.gz/microcore.gz files from the tinycore cd to a /boot directory on the hard drive, say /mnt/hda2/boot/. If you don't know where they are on the CD, use a command like "find /mnt/hdc -name tinycore.gz" and "find /mnt/hdc -name bzImage".
I am not sure the TC cd can boot from a kernel and initrd image on the hard drive, someone correct me if I am wrong. But using a boot disk like "super grub disk 2" will boot the TC installation. With the super grub disk, drop to a command line and enter commands like these at the grub> prompt:
insmod fat (if on fat32)
root (hd0,2) (if hda2 is where your files are)
linux /boot/bzImage quiet tce=hda2/tce home=hda2 (kernel image with boot codes)
initrd /boot/tinycore.gz (tinycore initrd file)
boot
This will let you do without any installed boot loader if there is that need, but it is obviously not the easiest way to use TC. And there are other ways besides the super grub disk, but I use super grub when dealing with a corrupt or missing boot loader.