"I don't see it as a point to prove. You let your actions speak for themselves," the bullish 60-year-old told reporters at his first news conference at the City Ground since being named as Nuno Espirito Santo's successor.
"You have to understand the responsibility you have. You have to provide joy to the supporters - that's why football clubs exist. They are the lifeblood of this game.
"That's all I care about. I'll win things here and people will still have doubts about me."
Postecoglou was axed by Tottenham Hotspur in June, days after winning the Europa League, a decision he knew was coming after the London club's 17th-placed finish in the Premier League.
"It wasn't great, I knew it was coming a fair way before the Europa League final," he said. "It's allowed me to move into this and maybe these things happen for a reason."
During a rare close-season off, Postecoglou did not know when, or if, another chance to manage in the Premier League would arise but on Saturday he returns to north London for his first game in charge, away to Arsenal.
"I was actually on the school run when the call came. It developed over the weekend but came to a head on the Monday," he said. "Once we'd got into discussions, it didn't take too much for me to realise it was a great opportunity.
"If you asked me at the start of the year if a job with Nottingham Forest was a possibility then I'd have said no because of the job Nuno had done."
Usually a new coach takes over a struggling team, but Espirito Santo put Forest back in Europe last season after a seventh-placed Premier League finish - 27 points better off than Postecoglou's Tottenham.
However, a falling out between Espirito Santo and Forest's owner Evangelos Marinakis, meant the Portuguese was doomed.
"The opportunity presented itself because of the work I've done," Postecoglou, who delivered Tottenham's first trophy since 2008 and twice won the Scottish title for Celtic, said.
"That's opened doors for me."
Postecoglou will need to win over Forest fans who will point to Tottenham's lamentable league season.
"They are not going to love me from the first day, I've got to earn their respect. It's a unique club where if you get that momentum, great things can happen," he said.
"Hopefully I can bring some special times to this club."
Postecoglou was criticised for sticking rigidly to his attacking style of play at Tottenham and said he would not make apologies for doing things his own way.
"I do like my teams to play exciting football and score goals. I want fans to be excited," he said. "But it is all with the end goal of winning football games."
With his Greek heritage, Postecoglou will at least have something in common with Marinakis whose relationship with Espirito Santo deteriorated so quickly.
"I love the fact that he's taken a club with a fantastic history but were in the Championship, into Europe. That tells you a lot about the kind of person he is," Postecoglou said.
Postecoglou faces the toughest of starts at Arsenal.
"It is what it is," he said. "First four games are away so I got a good look at the stadium today. Every Premier League fixture is challenging. It's a great game."