The general one-machine scheduling problem is strongly NP-Hard when the objective is to minimize the weighted number of late jobs. Few methods exist to solve this problem. In an other paper, we developed a Lagrangean relaxation algorithm which gives good results on many instances. However, there is still room for improvement, and a metaheuristic might lead to better results. In this paper, we decided to use a Genetic Algorithm (GA). Although a GA is somewhat easy to implement, many variations exist, and we tested some of them to design the best GA for our problem. Three different engines to evaluate the fitness of a chromosome are considered, together with four types of crossover operators and three types of mutation operators. An improved GA is also proposed by applying local search on solutions determined from the chromosome by the engine. Numerical experiments on different tests of instances are reported. They show that starting from an initial population already containing a good solution is very effective.