The Italian officially took over in April after a protracted takeover that initially saw him fail the Football League's owners' test.
A turbulent few months followed and the players were reportedly unhappy at being asked to return to the club during their summer break in May, while Brian McDermott was sacked as manager.
Cellino had also previously rejected reports that Ross McCormack would be sold, but the Scot eventually joined Fulham for around £11 million earlier this month.
Speaking to ITV Sport, the Leeds chairman said that he was acutely aware of the position he holds and pledged not to disappoint the club's loyal fanbase.
"I feel a lot of people are expecting a lot of me," he said. "I feel a lot of passion when the people ask me, and I fee a lot of responsibility for the people here to see something good.
"And I can't wait to make them happy, I don't want to make them disappointed."
A report from Sky Sports also quotes Cellino as claiming that he has rejected as many as four bids from other companies keen to take over at Elland Road.
He said: "I'm going to die [one day] but I'm going to do something good as well. You know how many people have come here asking to buy this club in the last 40 days? Four. Big companies.
"I told them I wouldn't sell for 100 billion. They said 'for one billion, you will.' I tried to explain to them, what is going to change if I sell for one billion?
"I don't have time to drink a glass of water so if I have one billion, so what? One kilo of caviar, 100 kilos of caviar... I'll work for my soul and the satisfaction of bringing this club back. How much is that worth? There's no price."