The Ecuadorian midfielder is on a year-long loan at Everton from Red Star Belgrade and he believes David Moyes' men have the capability to go all the way this term - starting with a victory on Thursday in the first leg of their first- round clash against Standard Liege.
He said: "I think we can win it, we are a great club and we have got a great manager."
He added: "I am very, very pleased to be part of this team because it is going places.
"It is up to the players to get our heads down, make it happen and take us forward.
"We have to get further than we did last year and it is up to us to make that happen."
Everton know all about their opponents after they narrowly lost to Liverpool in their Champions League qualifier last month.
Standard's 1-0 defeat at Anfield in the second leg of that fixture means they compete in the European football's "second" competition for the remainder of this season and Castillo is in no doubt they will pose a tough challenge.
However, he is still backing Everton to get past the Belgian side and he added he is expecting them to adopt some negative tactics to try and leave Goodison Park with a 0-0 draw at the very least.
He continued: "They will try and catch us on the break and we will have to be careful with how we go about things and how we get possession and how we use it. It will not be freeflowing because they will come here with a defensive job in mind."