Klaassen, a 24-year-old Netherlands international, has signed a five-year contract with the Toffees.
“I’ve walked around the training ground and talked with a few people and it feels good,” Klaassen said.
“I spoke a few weeks ago with Ronald Koeman about Everton. I wanted to come here and check if this was the right club for me and since I came here I feel so happy. It’s a great club.
“The moment that I spoke with the manager directly I got a good feeling. He’s an honest guy, we talked about the club, about the way he wants to play and that gave me a positive feeling.
“I talked to him about the club and everything we want to achieve and he was really positive. It’s difficult to leave Ajax but I think this is a good step for me now and I’m really looking forward to my time here.”
Earlier, Everton signed Pickford on a five-year deal. The 23-year-old said: “It’s great to sign for the club. Last season was my first in the Premier League so to get this opportunity with Everton now, to be able to go forward with my career and show everyone what I can do is unbelievable.”
Pickford has joined the Toffees for an initial £25m (€28.6m) which could rise to £30m (€34.3m) with add-ons, which would eclipse the £28m (€32m) deal the Merseysiders struck with Chelsea for striker Romelu Lukaku in 2014.
Having suffered relegation with Sunderland, Pickford is glad to be back in the top-flight so soon.
He said: “It’s a great club, a massive club and I think it’s a great opportunity and the right time for me to come to Everton and show what I can do.
“This is an exciting time for the club and for me, too, with the ambition being shown. I just want to be the best I can be to help.”
The fee is short of the £35m (€40m) Manchester City parted with earlier this summer to land Brazilian stopper Ederson — but still makes Pickford the most expensive British goalkeeper ever.
Elsewhere, Leicester have signed defender Harry Maguire from Hull for an undisclosed fee. The 24-year-old had a year remaining on his contract at Hull but following the club’s relegation from the Premier League he has joined the Foxes on a five-year deal.
A fee of around £17m (€19.4m) has reportedly been agreed between the two clubs and Maguire will link up with his new team-mates in early July ahead of Leicester’s pre-season training camp in Austria.
Meanwhile, Harry Redknapp has made “a good offer” to John Terry in a bid to bring the former England captain to Birmingham.
Terry will be a free agent on July 1 after ending his 22-year association with Chelsea at the end of last season.
A host of Premier League clubs have been linked with the 36-year-old, but Blues boss Redknapp hopes to tempt him into the Championship.
“I’d love to have him at the football club,” Redknapp said. “He’s a fantastic captain and great leader. We’ve made him a good offer and we have done the best we can. It’s up to John now, but we would love him at Birmingham.”