chern connection vs levi-civita connection

You need a non-Kahler complex manifold. Then the Chern connection will have nontrivial torsion. And the torsion corresponds to the non-closed Kahler form of the metric.

Further to a previous answer, you might like to refer to arXiv:0911.5655 and references therein for some interesting results on Chern connections on non-Kähler (and indeed non-integrable) almost-Hermitian manifolds.