The defender says they are focusing on becoming a better team under Jurgen Klopp.
The Reds take on Bournemouth in the League Cup on Wednesday evening, having narrowly edged past Carlisle United in the last round.
The competition has been blown open this season, with holders Chelsea knocked out by Stoke City and Arsenal being defeated by Sheffield Wednesday on Tuesday evening.
That offers Liverpool a great chance of pushing for major silverware this season, and Clyne says they are hoping to land a trophy for their loyal fan base.
“We set out trying to win every single tournament that we are in, so we want to try and get the win,” he told liverpoolfc.com.
“Hopefully we can go all of the way in the competition. We want to keep pushing to try and finish as high as we can in the league and hopefully we can bring home a trophy for the fans.”
The full-back arrived at Anfield from Southampton in the summer, and has quickly become the Reds' first-choice right-back under both Brendan Rodgers and now Jurgen Klopp.
It has been a difficult start to the season for his new side, with a number of disappointing results having ultimately cost Rodgers his job.
Klopp has now arrived and brought in his high-pressing style, but has so far failed to find a win in his first three games. Clyne, though, says his influence is noticeable already and they are working on becoming better as a collective.
“I’m settling in well. I’ve settled in with the players and also with the team,” he added.
“I think Jürgen Klopp is a good appointment and we are working on getting better as a team. You can see so far with the performances that we have done well, and we were just unfortunate not to get the three points at the weekend.”