Karius, who had previously been selected to start Liverpool's Champions League group stage games, was confirmed by Klopp as his first-choice stopper at a press conference on Friday.
The German was second-best in Premier League matches to Simon Mignolet before the club's win over Manchester City, and Karius told Welt that he is determined to stay in the Liverpool goal.
"I don't want to call it being number 1A or number 1B but the situation wasn't satisfying for me," the 24-year-old said. "Of course, I wanted to play every week. That's my aspiration. But I knew that I'd get a new chance. Now I want to take this chance.
"It's a professional relationship [with Mignolet]," Karius said when asked about the competition between the pair. "We train together and respect each other. I don't have any problem with him.
"In general I think that, nowadays, there are no such hostile relations between goalkeepers anymore. That's maybe different than in the past."
Liverpool signed Karius from Mainz 05 for £4.7m in 2016.