The Netherlands international was presented at the Nou Camp on Friday after sealing his move on a five-year deal for 13 million euros ($14 million).
Claudio Bravo, who moved to Manchester City on Thursday, played the club's league fixtures and Ter Stegen featured in the King's Cup and Champions League games after the pair joined Barca in 2014.
Ter Stegen is expected to be Luis Enrique's first choice in La Liga and the Champions League.
"The coach will decide but I'm going to make things tough for him," Cillessen told a news conference on Friday.
"Together me and Marc have to try and keep the coach guessing. It will be different to at Ajax but I'm sure that Barcelona will gain a lot.
"Marc is a great goalkeeper with marvellous qualities and I would like to be at the same level as him," added Cillessen on his new team mate and rival for the number one spot.
"I've come here to make things difficult for the coach but I'm not Ter Stegen's enemy, we're friends. I know that it's going to be difficult to play here because there's a lot of competition."
Cillessen is the fifth player to join Barcelona this summer, following Denis Suarez, Andre Gomes, Samuel Umtiti and Lucas Digne to the Nou Camp.
Barca sporting director Robert Fernandez revealed the club have one target remaining before the transfer window closes on Aug. 31, Valencia forward Paco Alcacer.
"It's Alcacer or no-one," Fernandez said on Friday.
"He's a player that interests us a lot and we are working towards it."
Barcelona confirmed that Brazilian full back Douglas had joined Sporting Gijon on loan for the season and midfielder and academy graduate Sergi Samper has moved to Granada on loan.
(Editing by Ed Osmond)
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