The existence of Kähler-Einstein metrics on a compact Kähler manifold of definite or vanishing first Chern class has been the subject of intense study over the last few decades, following Yau's solution to Calabi's conjecture. The Kähler-Ricci flow is a canonical deformation for Kähler metrics. In this expository note, we apply some known results of the Kähler-Ricci flow and give a metric classification for Kähler surfaces with semi-negative or positive first Chern class.