Eklund's parents were actors, but he initially decided not to follow in their footsteps and went to trade school. Around the age of nineteen, he was on a tour with his parents, where he made his stage debut. He subsequently began taking private acting lessons. He got a job at Helsingborgs city theater. In the 1950s, he was employed at Malmö theater, which was managed by Ingmar Bergman. Since then, he has performed with different theater troupes around Sweden.
Besides his substantial list of movie and TV credits, Eklund has dubbed many cartoons, including the Disney movies Bambi (Friend Owl), Lady and the Tramp, Aladdin (the sultan), The Great Mouse Detective (Flaversham), and others.