Watch the Movies
Gene was a performer who was not only successful in movies but also in Broadway. It is unlikely that you will find his Broadway performances on video but you can watch the movies in which he performed. This will provide you with a chance to see his good work and why he is admired so much. As a fan, you probably have done this already but it is not a bad idea that you see all his movies to have a complete look at his work.
There are going to be tons of reading materials available on Gene Kelly as he was a top notch Hollywood figure of his time. Read these and develop an insight about the life of your favourite artist.
Join a fan club which appreciates Gene Kelly. There are going to be many available both on and offline. You can try and be a part of one that is well known and followed online to be in touch with other fans. You can at the same time join a local club of Gene Kelly fans in your city of residence. This way you can stay connected with other official fans as well.
Keep an eye on events that are related to Gene Kelly. Every now and again, you will see an event that honours the great Hollywood performer and you can participate in it. It may not be possible to participate in events that are far away from where you reside but you can always drive to places nearby. Local fan clubs will almost always be celebrating events like birthdays for their star.