Well, first off, TinyCoreLinux isn't a turn-key solution as compared to Ubuntu Linux is. With that being said, in case you didn't read CGMINER's README file, here is a list of dependencies.
Dependencies:
curl dev library
(libcurl4-openssl-dev)
curses dev library
(libncurses5-dev or libpdcurses on WIN32)
pkg-config
libtool
AMD APP SDK (This sdk is mandatory for GPU mining)
AMD ADL SDK
(This sdk is mandatory for ATI GPU monitoring & clocking)
libudev headers
(This is only required for ASIC+FPGA auto-detection and is linux only)
libusb headers
(This is only required for ASIC+FPGA support)
Now, there is no ATI FGLRX or NVidia Detonator (restricted) drivers for this distribution, which puts you at a distinct disadvantage, you would have to do CPU mining only, such as Poolers CPU Miner (minerd).
This is the trick, you need to have a card that is capable of OpenCL, you need the restricted driver from the manufacturer and would work with this distribution of Linux. You need to download the correct version of the CGMINER, in this case, even if you have a 64-bit system and more than 4 GiB RAM you still need to use a 32-bit version of CGMINER because TinyCore and TinyCorePlus are both 32-bit.
Once you get past all of those hurdles, you have to make sure you have the correct version of gcc and supporting libraries to make and install all the required applications and libraries listed above.
After you've done that, then you can either download or have an existing configuration and run it for the first time.
Just a note, I haven't seen the correct version of gcc listed in the repositories here in TinyCore and besides, not having everything else you need, the libraries from AMD is a killer too.
If you really want to run something like CGMINER, I would get a normal Ubuntu Server edition and download it, assuming you're comfortable working in console / terminal. Because with the server edition you can load and install the drivers by command line and don't have to worry about X-Windows and the GUI sitting on top of that. Essentially you tweak and tune that server distro to remove unwanted / unneeded cron jobs and useless drivers from the start conf files.
While I don't like to shoo you or anyone else away from TinyCore and TinyCorePlus, this is not an operating system distribution of Linux for newbies and it's scant for the correct drivers you are looking for, that's the reason.
Another possibility is Slackware in console / text mode or a branch of Slackware in text mode called, VectorLinux.
I hope this helps you and sorry for the late response, this is the first time I saw this post.
-Alex