The talented winger completed a £25m switch from PSV on Wednesday.
And the 22-year-old wants to win the Champions League this season with his new club.
"It's an amazing move, amazing club. Whatever I can bring I will give for the club and for this team. It's a dream for every footballer to play in the Premier League; the atmosphere, the stadiums are always full," Bergwijn said at his unveiling on Wednesday.
"I want to score goals, give assists and win the Champions League. Last season I loved to see that you beat Ajax.
"When I was at training my father slept in the car and that's for three or four years. I know what they've [Bergwijn's parents] done for me and without them I wouldn't be here. I'm happy I can give them this back.
"When I was a young boy I looked to the coach, like 'oh it's Mourinho', now I play for him. You have to believe in yourself and I know what I can bring for this club. I'm good with the ball, I can score. They play good football and that's what I like. To play here is a dream for me. It's an honour."