The 21-year-old was bought for around £11m from Basel, and the Egyptian international has set his ambitions high after moving to Stamford Bridge.
Speaking to the club's official TV station, Salah said: "[To the Chelsea fans] I say, I hope we can win the Premier League and the Champions League.
"It is not easy, but this is a very good team and I hope we can win the league 100%.
"To play in this league is fantastic and it is my dream. As well as [playing in England's top division], I am also playing for one of the biggest teams."