The 23-year-old signed a four-year contract after Celtic succeeded with an improved, undisclosed bid.
Griffiths struck 28 goals for Hibernian last season during his second loan spell at his boyhood heroes, and hit 13 for Sky Bet League One promotion challengers Wolves this term, but some Celtic fans have questioned Neil Lennon's decision to sign him.
"Everybody is entitled to their own opinions," Griffiths said during his first media conference as a Celtic player.
"But it's down to the manager, he has brought me in, and it's up to me to show the fans what I can do.
"It's down to me to get my head down, work hard and go out there and show everybody that I can score goals and try and win everybody over.
"That's why you want to play football. You are always going to have doubters wherever you go.
"I've not got off to the best of starts with some Celtic fans, but hopefully I can win them over."
Griffiths has four Scotland caps and is untried at European level and he recognises he has to prove himself on the big stage.
"Of course I do," the former Livingston and Dundee forward said. "I have only played a handful of games for Scotland and I don't think I've done myself any justice so far.
"But there's no bigger place to play than the Champions League and hopefully in six months' time I will be doing that."
Lennon is convinced he can manage a player who has made headlines off the field, and is confident he will be rewarded.
"I think it has been somewhat over the top," Lennon said of the negative reaction from some fans.
"He is a young man. He has had a few off the field issues, as we all do. Sometimes they get blown out of proportion.
"So I will manage him as best I can and he has to concentrate on his football and if he does I am sure he will be a very good acquisition for the club.
"I have always liked his attitude on the field and having spoken to a lot of people who have played with him and worked with him, managers, they have spoken very highly of him."
Lennon added: "He is an engaging personality, he is quite quiet. Like most people who come with reputations he is not what you would expect.
"He seems quite a focused kid, he knows what he wants and he is delighted to be here. So we are very pleased.
"We think he is value for money. He's a prolific scorer, knows the environment here and I think we can make him better."
Griffiths only started two games for Wolves after being criticised by manager Kenny Jackett for his work-rate during a defeat by MK Dons on December 14, and he admitted he was desperate for the move to Celtic Park.
"It was in the middle of December that I first found out there was interest," he said. "As soon as the window opened, I was just praying for it to get done. It went to the last day, but it's finally done now and I'm delighted.
"I've left a very good club down at Wolves but I'm coming to the biggest club in Scotland, and a chance like that you can't turn down.
Griffiths had a strong rapport with his fellow Hibs fans, but the Edinburgh club could not agree a fee with Wolves to sign him on a permanent basis in the summer.
"If it wasn't Hibs I was coming to, it would only be Celtic (in Scotland). I will still be a Hibee as long as I live and it will be good to go back there in the coming seasons to play against them.
"But to play for Celtic is once-in-a-lifetime opportunity that you can't turn down."
Joe Ledley could be on his way out after Lennon confirmed the club had accepted a bid from Crystal Palace for the Wales midfielder.
Lennon also revealed he had bid for a player in Turkey, but the deal fell through when he could not agree terms with his present club.
Meanwhile, striker Mo Bangura is leaving Parkhead after failing to find the net in 16 games after signing in a £2.2million deal from AIK in August 2011.
Lennon said: "I am not sure when but he has been given a free, more or less."