In the Russian championships, Naumkin has twice been a bronze medallist. He achieved this feat in both the 1989 and 2001 editions of the championship. Naumkin has also contributed to team successes in chess. He was a member of the Moscow team that won the Russian team championship in 1999. Internationally, he represented Russia at the European Team Chess Championships in 1997 and 1999, scoring 7 points from 14 games.
Naumkin has been victorious in numerous other chess tournaments. These successes include winning the international tournament in Tsaghkadzor, Armenia in 1996, the open tournament in Vienna in 1997, the Paul Keres Memorial Tournament in Tallinn in 1998, and the round-robin tournament in Andorra in 1999.
In addition to his playing career, Naumkin is also a prominent chess trainer. He became a FIDE Senior Trainer in 2000.