Claudius Lysias was the name of the chief captain. Governor Felix was the Roman governor over the whole region. (See: [[rc://en/ta/man/translate/translate-names]])
This is a formal introduction to the letter. The chief captain begins by referring to himself. You can translate it in the first person. The words "am writing" are understood. Alternate translation: "I, Claudius Lysias, am writing to you, the most excellent Governor Felix, and I greet you" (See: [[rc://en/ta/man/translate/figs-123person]] and [[rc://en/ta/man/translate/figs-ellipsis]])