The Bees' eight-year stay in League Two was ended as their 2-0 defeat to Northampton left them in the relegation zone on goal difference.
And Kleanthous has told fans they can expect to see a number of homegrown players involved in the fight to make an instant return.
"Our focus moving forward will be to keep the spine of the team together and continue to integrate talented and hungry players from our academy into our first team," he told the club's official website.
"The academy has already produced the likes of Elliot Johnson, Luke Gambin, Mauro Vilhete, Jamal Lowe, Iffy Allen and George Sykes.
"This is just the beginning. Every day players within our youth ranks are growing in ability and desire that will fuel the club to new heights."
Next season will see Barnet playing away from Underhill, the club's home of 106 years. It was a move that had to be made, according to Kleanthous.
He added: "Ultimately it was impossible to sustain league football at Underhill and our relegation was unfortunately the culmination of a painful decline over the past four seasons.
"I wish things had worked out differently but we are where we are and next season we will begin life in the Conference Premier.
"However, that is already history. It is now time to look forward and we are already in the process of building for promotion."