Brendan Rodgers' side go into Sunday's clash with Swansea in fourth place in the Barclays Premier League, only four points adrift of leaders Chelsea.
This season has seen a remarkable turnaround from Suarez, who has put behind him a 10-match ban for biting Branislav Ivanovic and the summer's controversy where he stated his desire to leave Liverpool before changing his mind.
Suarez captained the team in Steven Gerrard's absence before signing a new contract in December, while he leads the Premier League scoring charts with 23 goals.
Hearing Gerrard talk about Liverpool's Champions League history has whetted Suarez's appetite to make a mark in the competition himself.
The Uruguayan said: "I watched the games on TV this week and when you hear that Champions League music that's motivation in itself.
"I want to be there with Liverpool. It's an unbelievable competition. It's where we want to be, playing against the top teams in Europe.
"Stevie has told me about Champions League nights at Anfield. He said when you have played Champions League football at Anfield you will never forget it. I want to try it for myself.
"When I arrived here I said it was my dream to play for Liverpool and play Champions League here. I know I can achieve my dreams here."
Liverpool have picked up 17 points from their last seven games to reignite their title challenge, but Suarez knows overhauling Chelsea, Arsenal and Manchester City will be tough.
He said: "Can we win the title? I'm not sure. I think this season the Premier League is so difficult. Every team can lose games. Man City, Chelsea and Arsenal have been dropping points.
"We are very strong at home and we must keep going like this. As the manager said, we are a chihuahua. We must just focus on every game. In football you never know what can happen."