You don't get a choice of the package manager. Generally speaking, each distribution only supports one low-level package manager, e.g.
dpkg with the deb package format on Debian/Ubuntu/Mint/…,
rpm with the RPM package format on Fedora/RHEL/SuSE/…, etc. For the most part each distribution has a single high-level package manager framework (APT on Debian, dnf on newer Fedora/RHEL, yum on older Fedora/RHEL, etc.) although occasionally you get a choice there.
If you have to use an unfamiliar package manager, the Pacman Rosetta can be useful.
You can use debootstrap to install Debian packages in a chroot. You can do this on any system with a Linux kernel, even Android. The Debian packages are installed in a separate hierarchy, not mixed with your existing operating system. Schroot automates most of the configuration. See the Ubuntu wiki or my tutorial for examples of schroot setup; they're for Debian/Ubuntu on top of Debian/Ubuntu but once you've installed schroot and debootstrap the setup is the same.
As schaiba remarked, “Core Linux” is probably actually Tiny Core Linux. It has its own package manager: Appbrowser, with an extensive available package list despite the small size of the basic installation. So you probably want to install the Tiny Core packages, not install APT.