The Uruguay striker had been in limbo since FIFA imposed a four-month ban from all football-related activity after his infamous bite on Italy's Giorgio Chiellini at the World Cup.
Suarez, who cannot feature for Barcelona in a competitive match until October 26, admitted he is also seeking professional help to avoid any further lengthy sanctions.
"The dream of playing for Barca comes from many years ago, as a kid. Everyone knows my relationship with family living here," he said.
"When I came to watch games here, I thought 'One day I will play here'. But it seemed something far away. There were offers from other clubs but when the Barca option came up I didn't think twice.
"My ban is what it is and we have to accept it. I have apologised and I've been talking to professionals in order to deal with it. I say to all the fans do not worry about that. I won't be doing that anymore.
"It's a difficult moment for me. I had the same situation last year at Liverpool (when he was banned for biting Branislav Ivanovic). I started to play on September 20 but I was able to train and help my team-mates off the pitch.
"I'm here to do important things at Barca. Everything that happens in the past, that needs to be forgotten. If I start thinking about everything I've gone through, I wouldn't be able to sleep."
After the Court of Arbitration for Sport eased some of the sanctions imposed by FIFA, allowing him to train with his new team-mates, Suarez came on for a late cameo in a friendly against Mexican side Leon on Monday.
"Playing at the Camp Nou yesterday was spectacular, very emotional. I still can't believe it. This is a dream come true," he added.
When asked if he felt any pressure over his £75m price tag, Suarez said: "Being a super star? The stars are in the sky. I'm here to help the team to try to win trophies."