The official bitcoin-0.8.0-linux.tar.gz has a directory structure to the archive. From the directory where you extracted it, you can run Bitcoin by typing
./bitcoin-0.8.0-linux/bin/32/bitcoin-qt & in a terminal.
You can also move the above bitcoin-qt file to another directory by itself, no other files from the archive are needed to run Bitcoin. You can be really fancy and
add an icon for it.
Make sure that you successfully created a persistent directory for your data, that you can shut down and reboot and your Bitcoin wallet addresses and the current block count are still there (just booting the live cd will always forget anything you downloaded as it only uses a RAM disk.) Also use UUID, so that partitions will always be found even if you use different USB ports or a different computer:
http://wiki.tinycorelinux.net/wiki:persistent_homenow that MSFT appears to be preventing booting from USB sticks in Windows 8, will your TinyCore USB system still work on those machines?
Newer machines may use UEFI boot instead of BIOS, this requires a 64 bit distro and UEFI bootloader (Ubuntu 12.10 64 bit is one distro that is ready, tinycore liveCD is not). This is not "Microsoft preventing booting", it is pretty standard for it to be enabled on new PCs now that Win8 supports it; old BIOS doesn't support hard drives larger than 2 TB. Macs also use EFI. Only
incompetent hardware manufacturers have a problem.
If "trusted boot" is enabled, a computer will only boot signed code.
This is being
worked on by kernel devs, but it requires kernel code signed by Microsoft. This is more of a grab by Microsoft that no user wants, under the guise of more security.
Both of these "features" can generally be disabled in BIOS. Windows 8 certified systems are required to permit the user to disable Secure Boot.