The Australian coach was dismissed earlier this month just 16 days after leading the club to their first silverware for 17 years and Levy explained that the club have aspirations to win the Premier League and Champions League that he wants to realise.
Speaking to Tottenham's in-house media team, the chairman said: "You saw the outpouring of emotion with the parade. It was just incredible. We've won a European trophy but it's not enough.
"It's what we haven't done that is more important, we need to win the league. We want to win the Premier League. We want to win the Champions League. We want to win."
Levy is "grateful" to Postecoglou for winning the Europa League but decided to replace him with Thomas Frank in order to give Spurs the best possible chance to be contenders in all competitions.
The Spurs chief said: "I'm very grateful to Ange. I don't regret appointing Ange. In his first season we finished fifth and in our second season we were over the moon to win the trophy.
"But we need to compete in all competitions and we felt that we needed a change. I've told him he's always going to be part of our history.
"It was a collective decision. It wasn't my decision. We do everything together. Emotionally it was difficult but we believe we've made the right decision for the club."
Levy is confident that former Brentford boss Frank can help the club progress after finally ending their trophy drought.
He said: "Whenever you have a new coach it's always a fresh start. You always have different ideas. But we want to build on the success of winning a trophy last season.
"One of the things that stood out to me with Thomas is that he's clearly highly intelligent. Great communicator. Super human being. Plus all the other technical aspects that are obviously important."