Finkel is the son of Rabbi Chaim Zev Finkel (1906–1965), founder of Yeshivas Heichal HaTorah in Tel Aviv and mashgiach at the Mir in Jerusalem. He is the grandson of Rabbi Eliezer Yehuda Finkel, who led the yeshiva in Poland and brought it to Israel. He is a cousin of Rabbi Nosson Tzvi Finkel, rosh yeshiva of the Mir from 1990 to 2011.
Finkel inherited his position as mashgiach of the Mir from his father. He began serving in this position under Rabbi Chaim Leib Shmuelevitz.