With three games to play last April, the Merseyside club were hot favourites to win the title for the first time since 1990.
Then they lost at home to Chelsea and only drew at Crystal Palace after leading 3-0, allowing Manchester City to come through and finish top by two points.
There was a further blow when top scorer Luis Suarez left for Barcelona, causing some critics to suggest Liverpool will not make the top four this season.
Rodgers said he was surprised by that, saying "People will rightly look to Manchester City and Chelsea but we are here to challenge."
He said that the big improvement last season after finishing seventh in his first season as manager will help this time, claiming: "What happened last year counts for nothing but the one thing that will carry on from that is the belief.
"We're entering now into a real long, hard season but a really exciting season because of the talent and the development of this group of players.
"Our aim is to be competitive. Like the top teams we want to win a trophy but it's about sustaining Champions League football. One you arrive in fourth you want to look beyond that and that would be our aim this season."
Liverpool are still hoping to sign "one or two more" players, Rodgers said.
They are understood to have agreed a fee for the Sevilla left-back Alberto Moreno.
He will not be available for Sunday's opening league game at home to Southampton, the club from whom Liverpool have signed three players in the transfer window.
One of them, England midfielder Adam Lallana, is injured but the others, Rickie Lambert and Dejan Lovren, are both available.