Can I use APT on TinyCore?

apt equlent of tinycore is tce-load http://wiki.tinycorelinux.net/wiki:package_management_cheat_sheet

You could probably compile dpkg/apt on TinyCore, but then you'd have to package stuff on your own.

But really, you are optimizing the wrong thing here. The major factors that define memory usage are the applications you use; you could probably do with Debian + a minimal window manager (e.g. Openbox). It should be lightweight enough.

Try tce-load -wi gcc.tcz for installing gcc. tce-load is the package manager and wi means download and install. Have a look at the list of packages available over at the TCZ repository.