Unto the most excellent governor Felix - The most honored, etc. This was a mere title of office.
Greeting - A term of salutation in an epistle wishing health, joy, and prosperity.

(13) Claudius Lysias unto the most excellent governor Felix [sendeth] greeting.
(13) Lysias is suddenly made by the Lord to be Paul's protector.

Claudius Lysias, unto the most excellent Governor Felix,.... This is the inscription of the letter, and by it we learn the name of the chief captain, so often spoken of in this and the two preceding chapters, which was Claudius Lysias; the first of these names is a Roman one, and which he might take from the Emperor Claudius, for he was not a Roman born; and the latter seems to be a Greek name, and was his proper name, and, he himself very likely was a Greek, since he purchased his freedom with money; one of this name was Archon of Athens (p); and another is reckoned by Cicero (q), among the famous orators of Greece, and is often cited by Harpocratian (r); one of Antiochus's noblemen, and who was of the blood royal, and acted as a general against the Jews, was of this name (s).
"So he left Lysias, a nobleman, and one of the blood royal, to oversee the affairs of the king from the river Euphrates unto the borders of Egypt:'' (1 Maccabees 3:32)
The chief captain calls Felix the governor
the most excellent, which was a title of honour that belonged to him as a governor; the same is given to Theophilus, Luke 1:3 sendeth greeting; or wishes all health and prosperity.
(p) Fabrieii Bibliograph. Antiqu. p. 213. (q) De Claris Orator. vel Brutus, c. 32. (r) Lex Decem Orator. (s) 1 Maccab. iii. 32.

Claudius--the Roman name he would take on purchasing his citizenship.
Lysias--his Greek family name.
the most excellent governor--an honorary title of office.