Studying on a class of planar quintic polynomial systems, the center and focus are judged by the formal series method. then by the Dulac function, the non-existence of closed orbit is discussed. And after Hopf bifurcation theory, some sufficient conditions for the existence of limit cycles of such systems are also obtained. Furthermore, with the theorem of Л.А.Черкас and Л.И.Жилевыч, some sufficient conditions for the uniqueness and stability of limit cycles of such systems are established .