The Belgium international joined los Colchoneros from Ajax in August, having reportedly spurned the advances of a number of interested parties, including Premier League side Liverpool.
He has since made eight appearances in all competitions for the club, with Atleti sitting second in La Liga, level on points with Barcelona, and securing a place in the knockout stages of the Champions League.
The 24-year-old told UEFA.com: "The adaptation process has been a good one and I am playing more and more so I am very happy.
"I proved to myself that I could do this and I proved to the coach that all I want is to do my best for this team. I am really enjoying playing here right now."
Atleti will be a darkhorse to reach the latter stages of European competition after drawing Serie A side AC Milan in the last-16, but Alderweireld is setting no limits on the club's progress.
"Towards the end of the group phase, even though we had qualified, we wanted to keep on winning and maintain that momentum. At this stage of the Champions League they are all big teams, so we'll see how we go."