We present an extensive experimental study of the conditions under which Cu forms encapsulated islands under the top surface layers of graphite, as a result of physical vapor deposition of Cu on argon-ion-bombarded graphite. When the substrate is held at 800 K during deposition, conditions are optimal for formation of encapsulated multilayer Cu islands. Deposition temperatures below 600 K favor adsorbed Cu clusters, while deposition temperatures above 800 K favor a different type of feature that is probably a single-layer intercalated Cu island. The multilayer Cu islands are characterized with respect to size and shape, thickness and continuity of the graphitic overlayer, relationship to graphite steps, and stability in air. The experimental techniques are scanning tunneling microscopy and X-ray photoelectron spectroscopy. We also present an extensive study using density functional theory to compare stabilities of a wide variety of configurations of Cu atoms, Cu clusters, and Cu layers on/under the graphite surface. The only configuration that is significantly more stable under the graphite surface than on top of it, is a single Cu atom. This analysis leads us to conclude that formation of encapsulated Cu islands is kinetically driven, rather than thermodynamically driven.