The Merseyside club came agonisingly close to winning the Premier League last term only for a late slump in form to hand the initiative to Manchester City.
After taking Liverpool from a seventh-place finish in his first season up to second 12 months later, Rodgers is now seeking a tangible reward for the improvements made on the pitch.
"Our expectancy this season is to win a trophy," he said. "That is what we'll be looking to achieve this year.
"My focus when I signed my first contract was purely to get us into the Champions League because we felt that'd put the club back where it belonged at that level and give us greater resources as a football club. We've achieved that goal.
"Now, as the philosophy is starting to become defined in the team, the next goal is for us to win a trophy.
"The most important thing for me as a coach is continuous improvement. It's very important that on the back of a good season where we arrived in the top four and just missed out on the title, we can go and push on again.
"Success for us is a moving target - we've got to keep moving, keep stretching ourselves and the expectancy of Liverpool is to win. That's what we'll look to do this season."
Liverpool have 18 top-flight titles, seven FA Cups and five top European crowns to their name, last winning silverware in 2012's League Cup, sealing a record-extending eighth triumph in the competition.
