The England defender was unveiled yesterday as Liverpool's first major signing of the summer, and said: "The boys did fantastically well last year. Hopefully we can now get a few more points and steal it from Manchester United.
"We want to go as far as we can in every competition but all the lads want to win the Premier League and that would be a great start.
"Liverpool went very close last term, a matter of just a few points. If we can nick a few more points this time, and cut out a few mistakes, then the title is very much on for us."
Johnson, who won a title medal at Chelsea after moving from West Ham as a teenager for £6million when the Hammers were relegated in 2003, now sees himself with another chance of a title chase.
He arrives from Portsmouth, after re-establishing himself as an England player, and added: "I had a spell at Chelsea before and there were good and bad times but when a club like Liverpool comes in for you it is not a hard decision.
"It is a fantastic club - one of the biggest in the world - with a great history and I wanted to be a part of that.
"The hairs on the back of your neck stand up when you come to Anfield and hear 'You'll Never Walk Alone' even when you don't play for Liverpool.
"So I cannot begin to imagine how it is going to feel when I put the kit on for the first time.
"We have got some fantastic players and I wanted to come here and help Liverpool win the league.
"I can definitely improve, every player can. I want to improve every aspect of my game.
"Rafael (Benitez) has told me he wants more of the same form I showed last year, but he also wants me to improve. We will be working hard together to do that, and I am here to do my best."
Benitez last night hailed Steven Gerrard's contract extension as "really good" for the future of the club after the Reds skipper signed his new deal this week.
Gerrard agreed the extension to keep him at Anfield until 2013 in April and has now completed the formalities after returning from his summer holiday to begin pre-season training.
Benitez has already secured the futures of Dirk Kuyt, Fernando Torres, Daniel Agger and Yossi Benayoun this summer and is delighted to have Gerrard signed up for the next four years.
He said: "We have spent this summer trying to secure the players we have.
"It's really good to keep Kuyt, Torres, Agger, Benayoun and Steven. They are the here and now of the club and for the future it is really good.
"We are really pleased. We have a lot of good players who want to stay, to win and compete.
"The message has to be the same again this year - to be there, to be contenders and to fight. If we can win trophies it would be fantastic."