MOSCOW—The Kremlin-backed leader of Russia’s southern Chechnya region announced that he wants to step down, as Russians commemorated the first anniversary of the murder of a prominent opposition leader, allegedly killed by a group of Chechens.

Ramzan Kadyrov, the 39-year-old leader of Chechnya, said in an interview with the Russian television network NTV that the time had come for the Russian government to find his replacement.

The...