“persecute” = dioko = to follow after, to pursue.  This is used in the sense of a hunter following after his quarry until he catches and kills it.

“had done” (KJV) = is actually in the continuous sense and should read “was doing” as it does in NIV. It could mean that Jesus had broken the Sabbath on other occasions or that with this incident He would go on to break the Sabbath on many more occasions.  In either case, it indicates a consistency in action; it was the habit of Jesus to “break” the Sabbath by doing good.

The Jews had disliked Jesus when He was first in Jerusalem (#2:18) and they were suspicious of his popularity #4:1.  But now he has broken the law and was even becoming a regular Sabbath-breaker. The result was that they began to hunt Him with the desire to kill Him.

John 2:18 Then the Jews demanded of him, “What miraculous sign can you show us to prove your authority to do all this?”

John 4:1 The Pharisees heard that Jesus was gaining and baptising more disciples than John,