Kántor started playing chess at the age of five. He won multiple national youth championships in Hungary. At age 12, he won the U12 section of the 2000 European Youth Chess Championship in Romania.
In 2004, he won the Zalakaros Open which helped him achieve the GM title at the age of 16 years and 11 months. He is one of the youngest chess players from Hungary to have achieved this.
In 2008, Kántor played for the Hungarian team that finished third in the World Youth U16 Chess Olympiad in Mersin, Turkey. In 2012, Kántor won the 1st Basel Chess Festival.
In addition, Kántor also studied law and has a legal career outside his professional chess. He graduated from Eötvös Loránd University Faculty of Law in Budapest, and subsequently received a two-year scholarship at the law faculty of the University of Hamburg, Germany. Afterward, he returned to Budapest and became an advocate.
Kántor has not been active in professional chess since 2015 due to his legal career. However, he participated in the 2016 European Club Cup in Greece and has stated that he is willing to participate in future Hungarian Championships if his schedule allows.