Capital punishment is not confined to the Old Testament and is not done to the exclusion of God’s grace, but rather as an act of God’s just judgment.
A simplistic view of vengeance involves the retribution of an evil deed. The first (Genesis 4:15) and last (Jude 7) mention of the word vengeance substantiates this definition. The Bible uses words like recompence (Deuteronomy 32:35), reward (Deuteronomy 32:41), punishments (Psalm 149:7), avenge (Jeremiah 46:10), revenge (Ezekiel 25:15), and repay (Romans 12:19) as descriptive words for vengeance. With this in mind, it is clear that vengeance is the just punishment as a consequence upon those who violate the laws of God. It is not the dispensing of wrath without cause but is specific punishment exacted for specific crimes committed.
