Finkelstein was born in 1951 and received an Orthodox Jewish upbringing. He passed the Israeli bar exam in 1976. He served in the Judge Advocate General's office in the IDF from 1973 to 2004, becoming a Major-General. He presided over the judgment that targeted killings were a legal part of the war on terror. In 2004 Hamas's Sheikh Yassin was killed under the policy. In July 2003 Finkelstein decided not to open a criminal investigation into the death International Solidarity Movement activist Rachel Corrie, but he did open an investigation into the death of Tom Hurndall in October 2003. In 2004 he testified before the Knesset's Law and Constitution committee and noted that "“In all the fatality cases we have investigated, there has never been a single incident of deliberate shooting [of civilians by the IDF]." He retired from the army in 2004 and became a district judge.