Kunte started playing chess at an early age and demonstrated a natural talent for the game. He soon began playing competitively on a national and then international level. His first big win was in 1985, when he won the World Under-8 Championship.
Kunte was awarded the International Master title in 1995 and the Grandmaster title in 2000 by the International Chess Federation (FIDE). His performances in several major tournaments established him as one of the top Indian chess players of his time.
Kunte has also represented India multiple times in the Chess Olympiad, an international competition of team chess. He competed in the 35th Chess Olympiad in 2002 and the 36th Chess Olympiad in 2004, among others.
In addition to his competitive career, Kunte is also an experienced chess coach. He has trained several young Indian players, including International Master Soumya Swaminathan and Grandmaster Vidit Santosh Gujrathi. He is the founder of Abhijit Kunte Chess Academy, a chess training center in Pune.