The Egypt international moved to Anfield from Roma in July in a deal that the Serie A club confirmed could cost the Reds as much as €50 million (A$75 million).
Salah, who is poised to make his competitive debut against Watford in the Premier League on Saturday, scored four times during an impressive pre-season campaign with his new side.
And the former Basel star is hoping Liverpool can challenge for trophies as the new campaign gets underway.
"It is a great club. It was a dream for me when I was young to play here one day," he told Sky Sports. "When I had the chance to come, I was thinking about whether I should stay in Rome or move here but because when I was young, it was a dream to me, I took the decision to come back.
"I would play as Liverpool on the PlayStation so I am happy to be here now. I was playing with a team like Steven Gerrard, Sami Hyypia and Jamie Carragher. I liked the team and the way they were playing. I was seven or eight years old so I was playing just to enjoy myself.
"I am very happy to be here and I would like to help the team so I am very excited to start in the Premier League games and I think we can win something this year together."
Salah joined Chelsea in 2014 but failed to settle at Stamford Bridge, and he was sent on loan to Fiorentina and Roma before joining the latter permanently last year.
The 25-year-old believes he is a much better player than he was when Jose Mourinho brought him to England the first time and he is determined to prove his doubters wrong.
"It was always in my mind to come back. I like the challenge, I like English football and I want success here," he said.
"I was not playing at Chelsea so I took the decision to go somewhere else. I went to Fiorentina and did six months and with Roma, I was there for two seasons so it was a good challenge for me. I had a good time with Roma and I did well so it was a good time to come back.
"As a person, as a man, even on the pitch you can see that I have a lot of experience now. I played in Italy, I was in Switzerland, I came to England then I went back to Italy so it's a big change in my life.
"I think I have improved everything. I think now, compared to three or four years ago, I can defend more, I can play more tactically, I can finish better than I could before. I scored about 40 goals with Roma so that's quite good. I always like to improve myself."