About that time (kat' ekeinon ton kairon). Same phrase in Rom 9:9. That is, the early part of a.d. 44 since that is the date of Herod’s death. As already suggested, Barnabas and Saul came down from Antioch to Jerusalem after the persecution by Herod at the end of 44 or the beginning of 45.
Herod the king (Hērōidēs ho basileus). Accurate title at this particular time. Herod Agrippa I, grandson of Herod the Great, was King of Palestine a.d. 42 to 44; only for these three years was a Herod king over Palestine since the death of Herod the Great and never afterwards. Archelaus never actually became king though he had the popular title at first (Mat 2:22).
Put forth his hands (epebalen tas cheiras). Second aorist active indicative of epiballō, old verb, to cast upon or against. The same idiom with tas cheiras (the hands, common Greek idiom with article rather than possessive pronoun) in Act 4:3; Act 5:18.
To afflict (kakōsai). First aorist active infinitive of kakoō, old word to do harm or evil to (kakos), already in Act 7:6, Act 7:19. Outside of Acts in the N.T. only 1Pe 5:13. Infinitive of purpose. Probably the first who were afflicted were scourged or imprisoned, not put to death. It had been eight years or more since the persecution over the death of Stephen ceased with the conversion of Saul. But the disciples were not popular in Jerusalem with either Sadducees or Pharisees. The overtures to the Gentiles in Caesarea and Antioch may have stirred up the Pharisees afresh (cf. Act 6:14). Herod Agrippa I was an Idumean through his grandfather Herod the Great and a grandson of Mariamne the Maccabean princess. He was a favourite of Caligula the Roman Emperor and was anxious to placate his Jewish subjects while retaining the favour of the Romans. So he built theatres and held games for the Romans and Greeks and slew the Christians to please the Jews. Josephus (Ant. XIX. 7, 3) calls him a pleasant vain man scrupulously observing Jewish rites. Here we have for the first time political power (after Pilate) used against the disciples.