Born in Chicago in 1943, Finkel was a descendant of a rabbinic dynasty and was named after the famed Alter Slabodka, a leader in the mussar (ethics) movement. During his 21-year tenure at Mir Yeshiva, student attendance rose dramatically, making it the biggest yeshiva in Israel with enrollment around 6,000. It was announced at the funeral that Finkel's son Rabbi Eliezer Yehuda would succeed his father as the yeshiva head. The rabbi had suffered from Parkinson's disease for over 30 years before his death but continued to give Torah lectures and travel for fundraising efforts. The day before his death, Finkel had attended the funeral of another yeshiva head, Rabbi Dov Schwartzman. The funeral procession began at the yeshiva and led to the Har Hamenuhot cemetery in Givat Shaul.