Leicester won the English Premier League (EPL) by 10 points last season, deposing 2015 champions Chelsea. And after the first top-flight title in their history, King is eager for more.
"Once you get a taste of silverware like we did last year, we almost want to get as much as possible," he said on Monday.
"We know it doesn't always work like that but we will be going in to obviously try and get as far as we can."
"It is a tough game and I am sure they are going to want to come pretty strong as they are not in any European competitions this year. They are only playing weekends in the league. It is going to be a tough game."
"We're going to have to go into it and be at our best as we want to go as far as we can in every competition."
Chelsea's 1-2 defeat at Leicester in December last year cost coach Jose Mourinho his job and Italian Antonio Conte was named his permanent replacement before the start of the current season.
Conte lost his first game as the Chelsea coach in Friday's 1-2 loss against Liverpool but forward Eden Hazard believes Conte will be a success.
"He's a top manager, he's a top man," he told Chelsea TV on Monday.
"He has the passion to do his job. He gives everything on the bench and we try to give everything on the pitch.
"He tries to speak a lot with the players, to talk a lot, and we are pleased to work with him."
