After signing a four-year deal to succeed Manchester United-bound David Moyes, the new Toffees manager is aiming to break the top four after building on the Scot's foundation.
The former Wigan manager was appointed as Manchester United-bound David Moyes' successor on Wednesday, signing a four-year deal.
"When David Moyes first came to see me 11 years ago, we were in a bad state. His first words to me were 'You're not going down'," Kenwright told reporters at Martinez's official unveiling. "Almost Roberto's first words to me were 'I'll get you in the Champions League'.
"That is extraordinary from a man who does not know much about the football club from the inside, but has obviously studied the football club and looked at what were were doing for the last few years."
Martinez backed up the claim, adding that Moyes' achievements over the previous 11 years has set the club up to battle with the Champions League-chasing pack and outlined his plan ahead of the new season.
"The chairman hasn't put any pressure on me, says he wants to get in the Champions League!" Martinez joked. "I think it's important. The fans need to know that they are the most important part of this football club and we will need them to lead us.
"We've got a very strong team - I think what David Moyes has done in this football club is set a real high standard - and over the next few months we need to get together and make sure we are a winning side and a very strong side that can carry on looking to achieve more.
"I think finishing in the top six was a magnificent achievement for Everton, and there's many things we need to carry on building if we want to improve that."
Kenwright insisted he would back Martinez in the transfer market, and confirmed Marouane Fellaini's buyout clause, but insisted that the rest of the squad is fully behind the Spaniard's appointment.
"I'll give him a few quid to sign a few players," the chairman continued. "We all know there is a player who has got an escape clause, but the others are not for sale."