He was born in 1975 in Pančevo. He graduated from the Kornelije Stanković High School of Music in Belgrade, after which he continued his education at the Georges Bizet Conservatory in Paris. At the National Conservatoire in Paris, he graduated with the highest grades. Two years later, he obtained his master’s degree at the Ecole Normale in Paris (École normale supérieure), and in 2008. he also received his master’s degree at the Academy of Arts in Novi Sad. 

Klenkovski is the laureate of numerous competitions in the country and abroad, the most important of which are the first prizes at the International Competition Leopold Belan in Paris, the International Competition in Picardy, the Federal Competition in Ljubljana, as well as the First special prize at the International Competition in Paris and the Third prize at the International Competition Musical Youth in Belgrade. 

He performed as a soloist with the Royal Orchestra from Brussels, the Philharmonic of Nations from Germany, the Belgrade Philharmonic, the RTS Orchestra, the Symphony Orchestra from Niš, the Belgrade Ensemble Dušan Skovran, the String Orchestra of St. George, etc. 

He has also performed in many chamber ensembles in the country and abroad, with special guest appearances in Spain, Russia, China, Germany, France, Slovenia, Austria, Switzerland, Italy, Greece, Slovakia, Czech Republic, America and India. Until 2002, Veljko Klenkovski performed as the first clarinetist of the Philharmonic Orchestra in Germany, as well as the first clarinetist of the National Theatre Opera Orchestra in Belgrade, the Madlenianum Chamber Opera and the RTS Orchestra in Belgrade. 

Since 2002, he has been the solo clarinetist of the Belgrade Philharmonic. Also, since 2015, he is the solo clarinetist of the National Theatre in Belgrade. He is the founder and member of the Belgrade Philharmonic Wind Quintet, as well as the Belgrade Wind Trio. 

He performed with the Bruno Walter Symphony Orchestra, the Harbin Symphony Orchestra (China), as well as the Cyprus National Orchestra. During his many years of work in the orchestra, he collaborated with many eminent musicians, including Zubin Mehta, Christian Mandeal, Barry Douglas, Muhai Tang, Leopold Hager, Paul Angerer and others. Klenkovski is also actively engaged in pedagogical work. He held master classes in France, Serbia, Slovenia, Montenegro and North Macedonia. 

He is a member of the jury of the international competition “Pearls of Ohrid” in Ohrid and “Saxclar” in Skopje (North Macedonia), Davorin Jenko-Belgrade and in Požarevac (Serbia). 

He is the winner of the Belgrade October Award and the award for pedagogical work. He received his doctorate in 2015 at the Faculty of Music in Belgrade and was awarded the Bruno Brun award, as a student who completed his doctoral studies with the best results. Since 2016, he has been working as an assistant professor at the Faculty of Music in Belgrade.

Veljko Klenkovski is a Buffet Crampon artist and plays Legend brand clarinets.