Despite Maurizio Sarri's summer departure, he had a successful campaign, finishing third in the Premier League and winning the Europa League, meaning Lampard has a tough act to follow.
Christensen told Sky Sports: "At Chelsea, you have to have high expectations. We are a club that are used to winning and that hasn't changed. We don't want to change and we're aiming for the highest we can every season.
"We don't know what that means yet this season but we are definitely going to push to win something, that's what we want and that's what Chelsea's about.
"The manager has won the Premier League and now there are a lot of young players who are hungry to do it. That mixture, at the minute, is very healthy and I think it's something the club needs. We're pushing in training to be the best we can this season."