Designing general binary self-orthogonal codes is a difficult problem in both classical coding theory and quantum coding theory. The structure of one-generator quasi-cyclic codes constructed by concatenating binary circulant matrices is investigated. Twenty-eight optimal or best known binary self-orthogonal codes are built by designing the structure of a special subclass of quasi-cyclic codes, which takes advantage of some restrictions such as the shifting equivalence relation on vector, the equivalence relation on linear codes and even weight property of binary self-orthogonal codes. A puncturing-expurgating construction method for binary self-orthogonal codes is proposed, and sixty-two derived codes from these obtained self-orthogonal codes are constructed. In comparison with Literature (13), 67 and 23 among our ninety self-orthogonal codes are separately optimal and best known. The construction results indicate that these two methods are effective to design general self-orthogonal codes. Furthermore, the ideas can preferably solve the construction problem of self-orthogonal codes with possible larger minimum dual weight, which is the critical infrastructure in designing better quantum codes.