We consider the Hamiltonian and Lagrangian formalism describing free κ-relativistic particles with their four-momenta constrained to the κ-deformed mass shell. We study the formalism with commuting as well as noncommuting (i.e., with nonvanishing Poisson brackets) space-time coordinates; in particular a κ-deformed phase space formalism leading to the κ-deformed covariant Heisenberg algebra is presented. We also describe the dependence of the formalism on the various definitions of the energy operator corresponding to different choices of basic generators in the κ-deformed Poincaré algebra. The quantum mechanics of free κ-relativistic particles and of the free κ-relativistic oscillator are also presented. It is shown that the κ-relativistic oscillator describes a quantum statistical ensemble with a finite value of the Hagedorn temperature. The relation to a κ-deformed Schrödinger quantum mechanics in which the time derivative is replaced by a finite difference is also discussed.