Meshkini was an Iranian Azerbaijani born in a village near Meshkin Shahr and the Sabalan mountain. He was born as Āli, but preferred the pronunciation Ali during his career. He succeeded Ayatollah Montazeri as chairman of the Iranian Assembly of Experts after the latter was ousted. The Assembly of Experts selects the Supreme Leader of Iran and supervises his activities. Meshkini chaired the commission that drafted the amendment of 1989 to the 1979 constitution of Iran. Within the commission, he also chaired one of its four committees, the one tasked with drafting amendments that would strengthen the judiciary. As such, he was an influential force in changing the leadership of the judiciary from a triumvirate, the Supreme Judicial Council, to an individual in 1989.
He was the author of many books on Islamic jurisprudence and general issues of Islam. Mohammadi Reyshahri, Iran's Minister of Intelligence from 1984–1989, was Meshkini's son in law.
Meshkini was the chair of Society of Seminary Teachers of Qom and also served as Friday Prayer Leader in Qom. Despite his old age and poor health, he became a candidate for re-election in 2006 Assembly of Experts elections.