James joined the Merseysiders from Real Madrid on Monday as the Colombia midfielder reunited with Toffees manager Carlo Ancelotti.
The 29-year-old, who previously worked with Ancelotti at Real and Bayern Munich, is the most high-profile of the manager’s close-season signings and his arrival has raised expectations at Goodison Park.
Having won La Liga and Champions League titles with Real and enjoyed domestic success in Munich, James believes Everton can lift major silverware for the first time since the 1995 FA Cup.
“Why not? It is a club that means business,” James told a press conference on Thursday.
“There are a lot of serious-minded people at the club at all levels who mean to achieve and that means winning trophies.
“That won’t happen overnight but I have seen real positive signs we can make progress. I think trophies are more than a possibility.
“I am the sort of person that always wants to achieve more and more in my career and my life.”
Rodriguez arrived on Merseyside to a huge fanfare, with the club projecting his image onto buildings in Colombia, New York and Miami Beach, in addition to Goodison Park and the Royal Albert Dock in Liverpool.
James is expected to live up to the hype at a club starved of silverware and forever stuck in the shadow of their more successful neighbours Liverpool.
“It is great for the club and nice for me also that there is so much interest but I don’t pay so much attention,” James said of the pressure to perform.
“I am here to play football and focus on my game. I am here to learn, focus on what I need to do and play well.
“I have seen the fans are proper fans, full of passion and when players see the fans are so passionate we want to give even more to make them happy.”
Ancelotti is confident he can bring the best out of the 2014 World Cup Golden Boot winner who fell out of favour at Real.
“We can expect him to show his quality. He has a lot of quality and I am sure he is going to show it in the Premier League,” said the Everton boss, whose side travel to Tottenham on Sunday.
“It is a fantastic competition and he deserves to play in it. He has experience in different leagues and I am sure he is going to show the quality he showed in different countries.”