Was Luke a Gentile? Maybe a proselyte? If so, how do we know this?
Nothing is said in the Scriptures concerning Luke's heritage, so we cannot know by direct reference.
Luke is believed to be a Gentile because in Colossians 4:10-14, Paul lists three fellow Jews and then three others, of which one is Luke. There is another hint in Acts 1:19, "And it became known to all those dwelling in Jerusalem; so that field is called in their own language, Akel Dama, that is, Field of Blood." Though Akel Dama is in Aramaic, Luke calls it "their own language" and not his own.
There is one instance that leads me to believe Luke was of the Jewish faith before he became a Christian. Paul had traveled to Jerusalem along with Luke (seen by the "we" in the passage). "After these days we took up our baggage and went up to Jerusalem. Some of the disciples from Caesarea also went with us, bringing one Mnason of Cyprus, an early disciple, with whom we would stay. When we had come to Jerusalem, the brothers received us gladly" (Acts 21:15-17). It was suggested that Paul bring four men to the temple for their purification rites so that Jews would see that Paul wasn't hostile to the Old Law. But the idea backfired. "When the seven days were almost completed, the Jews from Asia, when they saw him in the temple, stirred up all the multitude and laid hands on him, crying out, "Men of Israel, help! This is the man who teaches all men everywhere against the people, and the law, and this place. Moreover, he also brought Greeks into the temple, and has defiled this holy place!" For they had seen Trophimus, the Ephesian, with him in the city, and they supposed that Paul had brought him into the temple" (Acts 21:27-29). Though Luke was a well-known companion of Paul, it was the Gentile Trophimus who got the crowd riled, assuming he came with Paul into the Temple (though he had not been there).
Thus by inference, we suppose that Luke was possibly a Hellenistic Jew or more probably a proselyte.