Self-orthogonal codes are a subclass of linear codes that are contained within their dual codes. Since self-orthogonal codes are widely used in quantum codes, lattice theory and linear complementary dual (LCD) codes, they have received continuous attention and research. In this paper, we construct a class of self-orthogonal codes by using the defining-set approach, and determine their explicit weight distributions and the parameters of their duals. Some considered codes are optimal according to the tables of best codes known maintained at \cite{Grassl} and a class of almost maximum distance separable (AMDS) codes from their duals are obtained. As applications, we obtain a class of new quantum codes, which are MDS or AMDS according to the quantum Singleton bound under certain conditions. Some examples show that the constructed quantum codes have the better parameters than known ones maintained at \cite{Bierbrauer}. Furthermore, a new class of LCD codes are given, which are almost optimal according to the sphere packing bound.