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We present \textsc{C}lassification of \textsc{C}luster \textsc{Ga}laxy
\textsc{Me}mbers (\textsc{C$^2$-GaMe}), a classification algorithm based on a
suite of machine learning models that differentiates galaxies into orbiting,
infalling, and background (interloper) populations, using phase space
information as input. We train and test \textsc{C$^2$-GaMe} with the galaxies
from UniverseMachine mock catalog based on Multi-Dark Planck 2 N-body
simulations. We show that probabilistic classification is superior to
deterministic classification in estimating the physical properties of clusters,
including density profiles and velocity dispersion. We propose a set of
estimators to get an unbiased estimation of cluster properties. We demonstrate
that \textsc{C$^2$-GaMe} can recover the distribution of orbiting and infalling
galaxies' position and velocity distribution with $<1\%$ statistical error when
using probabilistic predictions in the presence of interlopers in the projected
phase space. Additionally, we demonstrate the robustness of trained models by
applying them to a different simulation. Finally, adding a specific star
formation rate and the ratio of the galaxy's halo mass to the cluster's halo
mass as additional features improves the classification performance. We discuss
potential applications of this technique to enhance cluster cosmology and
galaxy quenching.