When a compact Kahler manifold satisfies $c_1=0$, it admits a Ricci-flat Kahler metric by Calabi-Yau, hence its tangent bundle is semistable. Then its discriminant $\int_M 2nc_2\wedge \omega^{n-2}$ is non-negative, by Bogomolov inequality, and positive when the curvature of $TM$ is non-zero. Therefore, a compact Kahler manifold with $c_1, c_2=0$ admits a flat Kahler metric, hence by Bieberbach's solution of Hilbert XVIII it is a quotient of a compact torus by a finite group which acts freely.

When $M$ is non-Kahler, this cannot be applied, and there are many examples of complex manifolds with vanishing Chern classes, such as Hopf manifolds, Calabi-Eckmann manifolds, Inoue surfaces, complex nilmanifolds, holomorphic parallelizable manifolds, $SL(2,{\Bbb C})/\Gamma$ which Ben McKay mentioned, and so on, and so forth.

When a compact Kahler manifold satisfies $c_1=0$, it admits a Ricci-flat Kahler metric by Calabi-Yau, hence its tangent bundle is polystable (direct sum of stable bundles of the same slope). Then its discriminant $\int_M 2nc_2\wedge \omega^{n-2}$ is non-negative, by Bogomolov inequality, and positive when the curvature of $TM$ is non-zero. Therefore, a compact Kahler manifold with $c_1, c_2=0$ admits a flat Kahler metric, hence by Bieberbach's solution of Hilbert XVIII it is a quotient of a compact torus by a finite group which acts freely.

When $M$ is non-Kahler, this cannot be applied, and there are many examples of complex manifolds with vanishing Chern classes, such as Hopf manifolds, Calabi-Eckmann manifolds, Inoue surfaces, complex nilmanifolds, holomorphic parallelizable manifolds, $SL(2,{\Bbb C})/\Gamma$ which Ben McKay mentioned, and so on, and so forth.