Hacham Abraham Hacham was born to Lean and Hacham Azaria in Sanandaj, Kurdistan, in 1882.
He began his Torah studies with his father, Hacham Azaria, who was one of the city's sages and rabbinic judges. When his father died, Hacham Abraham Hacham's brother was appointed the city's rabbi. When his brother died as well, Hacham Abraham Hacham adopted his brother's sons and replaced him as the city's rabbi.
In 1912 he moved to the city of Kermanshah, where there were some six thousand Jews at the time, to officiate as its rabbi. Hacham Abraham Hacham officiated in all the rabbinic roles. He was also responsible for Jewish education at the Kol Israel Haverim – Alliance school, attended by many of the city's Jewish children. As part of his position, he developed good relations with the city's governor and with its Muslim sages. These relations served him to help the Jews of his city and of other cities as well.
Hacham Abraham Hacham harbored a fierce love for the Holy Land and, on of Passover Eve of 5696 (1936), immigrated to the Land of Israel, settling in the Jerusalem's Bucharim neighborhood. In 1942 he began to officiate as Rabbi of the Beit Hannan moshav; after a year's time he moved to Rishon LeZion and officiated as Rabbi of the community of Urfa Jews. When the State was founded, he returned to Jerusalem and established the Tipheret Jerusalem synagogue and the Magen Abraham yeshiva, serving as the rabbi, preacher and spiritual leader of both institutions.
A few days before his demise, on a Sabbath Eve, he told his wife that he could not see his reflection in the Kiddush cup and that he sensed his end approaching. At midnight of the 4th of Elul, 5724 (1964), he rose from his bed, washed his hands and kneeled at the entrance of his room beneath the mezuzah. When his wife approached, he requested a towel. After drying his hands, he recited the appropriate blessing and, as he leaned his head on his wife's shoulder, his soul left his body in sanctity and purity.
His only book is Sepher Izraeli – a collection of sermons and commentary on all the Torah Readings. He spent most of his life writing and editing the book, but submitted it for publication only two months before his death.