The Colombia star joined Los Blancos this summer in an €80 million (£65m) deal from Monaco and has already had successes and failures in his time at the club.
James played in his side's Uefa Super Cup victory over Sevilla and scored against rivals Atletico Madrid in the Supecopa de Espana, though Carlo Ancelotti's men went on to lose 2-1 on aggregate to Diego Simeone's team.
However, the 23-year-old has now set his sights firmly on winning the continent's greatest prize as he works to improve alongside the likes of Cristiano Ronaldo and Gareth Bale.
"I think it's something great to win the Champions League with Madrid, it's something special to be at a club that wants to win something and I hope we can win it again," he told reporters at the unveiling of Madrid's new Champions League strip.
"It's a club from another world, unique. To be alongside great stars is something really special. I hope I can get better, bit by bit."
James made his league debut for Madrid in the 2-0 win over Cordoba at the Santiago Bernabeu on Monday.