Ablowitz and Musslimani proposed some new nonlocal nonlinear integrable equations including the nonlocal integrable nonlinear Schrödinger equation. In this paper, we investigate the Darboux transformation of coupled nonlocal nonlinear Schrödinger (CNNLS) equation with a spectral problem. Starting from a special Lax pairs, the CNNLS equation is constructed. Then, we obtain the one-, two- and $N$-soliton solution formulas of the CNNLS equation with $N$-fold Darboux transformation. Based on the obtained solutions, the propagation and interaction structures of these multi-solitons are shown, the evolution structures of the one-dark and one-bright solitons are exhibited with $N$ = 1, and the overtaking elastic interactions among the two-dark and two-bright solitons are considered with $N$ = 2. The obtained results are different from those of the solutions of the local nonlinear equations. Some different propagation phenomena can also be produced through manipulating multi-soliton waves. The results in this paper might be helpful for understanding some physical phenomena described in plasmas.