In this paper, a numerical method is introduced to find the eigenvalues and eigenfunctions of the Caputo fractional Dirac operator. To this end, the problem reduces to a Volterra integral equation with a weakly singular kernel. Then, the pseudospectral method based on Chebyshev cardinal functions is used to solve the obtained Volterra integral equation. By introducing the operational matrix of the fractional integral operator for cardinal Chebyshev functions, the Volterra integral equation is reduced to an algebraic system. To obtain the approximation of the eigenvalues, it is sufficient to find the roots of the characteristic function of the algebraic system. Then, the convergence of the method is proved. To demonstrate the ability and accuracy of the method some numerical examples are solved.