In this paper, we aim to investigate the spectrum of the nonselfadjoint operator L generated in the Hilbert space l2(N, C2) by the discrete Dirac system{yn+1(2)−yn(2)+pnyn(1)=λyn(1),−yn(1)+yn−1(1)+qnyn(2)=λyn(2),n∈N, and the general boundary condition∑n=0∞hnyn=0, where λ is a spectral parameter, Δ is the forward difference operator, (hn) is a complex vector sequence such that hn=(hn(1),hn(2)), where hn(i)∈l1(N)∩l2(N), i= 1 , 2 , and h0(1)≠0. Upon determining the sets of eigenvalues and spectral singularities of L, we prove that, under certain conditions, L has a finite number of eigenvalues and spectral singularities with finite multiplicity.