Who was in charge of building the missions? Who built San Luis Rey?

The answer is a little complex because you asked a very general question. The ultimate decision to settle Alta California and build a chain of missions was made by the King of Spain. The king's representative in New Spain was a Viceroy, located in Mexico City who organized the effort. The actual building of the missions was the joint responsibility of the Governor of Alta and Baja California and the Father President of the Mission Chain (who was initially Junipero Serra). The Father President was involved in selecting each site. The actual building of a given mission was the responsibility of the two missionaries assigned to that mission.

San Luis Rey was built under the overall direction of Father President Lausen. The lead missionary was Father Peyri, who served at the mission for 34 years.