The Toffees have spent £20 million on the Colombian attacking midfielder - who will team up with manager Carlo Ancelotti for the third time in his career.
Rodriguez, who won the Golden Boot at the World Cup in 2014, said: "I am really, really happy to be at this great club, a club with so much history, and here with a manager who knows me really well.
"I'm looking forward to achieving great things here - and winning things, which is what everyone's aim is.
"I've come here to try to improve, to get better. I've also come here to help the team to win, to play good football - entertaining football."
James previously worked with Ancelotti at Real Madrid and Bayern Munich and admits that the Italian coach was a big reason behind his decision to join Everton.
The 29-year-old player said: "I'm convinced, with Carlo and his technical staff, we can achieve big things and one of the big reasons [I signed] was the presence of Carlo Ancelotti.
"I have enjoyed some great times with him previously at two different clubs. That was a massive reason to come here."
Ancelotti commented: "James, as everyone knows, is a fantastic player with a lot of quality and a lot of ability to make assists for the strikers.
"He's really excited about the project of the club and the fact we want to improve. He was really excited to sign and I didn't spend time to convince him - he was already convinced."