The 29-year-old has won the Champions League, La Liga and Copa del Rey with Barcelona since joining them in 2014.
But prior to the move he was suspended for four months for biting Italy's Giorgio Chiellini at the World Cup.
"I know they'll remember the bad things I've done and I can't change that but I want to be remembered for the good things," he told the Daily Mail.
Suarez joined Liverpool in January 2011 and had already been banned for biting whilst playing for Ajax.
The Uruguayan won a host of individual honours during his three-year Anfield career, but also received a 10-match ban for biting Chelsea's Branislav Ivanovic and an eight-match suspension for racially abusing Manchester United's Patrice Evra.
However, in his first year with Barcelona he was on the scoresheet in the Champions League final as his new team beat Juventus 3-1 in Berlin.
"I went from being the bad guy, the worst player who gets all the criticism, to being the player who makes the difference and scores the important goals," he said.
"There is always a chance to get even and I'm proud of the fact that for all the blows I always got back up again."
Asked about his future, Suarez said: "I'd prefer to stay here [at Barca] for many more years. I know it doesn't always turn out that way. But if I had to return to the Premier League, I would only go to Liverpool."
Barcelona and former Liverpool striker Luis Suarez wants to be known for his football rather than his controversies.