Born in Dale, Texas, Weldon began his career as a disc jockey at KWCO in Chickasha, Oklahoma in 1946, remaining there for four years. He made his television debut in 1950, as "Uncle" Jimmy (himself) and the voice of the duck puppet, Webster Webfoot on "The Webster Webfoot Show", produced locally by Dallas station WFAA-TV.
Weldon moved to California in the early 1950s and continued to make television appearances while in Hollywood. In 1959, he and Webster Webfoot returned to the air again as the hosts of "HI, MOM" on NBC affiliate WRCA-TV after Shari Lewis left. They would continue performing their show on KJEO-TV in Fresno throughout the 1960s.
It was thanks to Weldon's "quacky" voice that he earned the voice-over for the Hanna-Barbera cartoon character, Yakky Doodle Duck. Yakky Doodle made frequent appearances on the Yogi Bear cartoons, and then got his own show, "The Yakky Doodle Duck Show", from 1960 to 1962.
Weldon has also made numerous television appearances in acting on shows such as Dragnet, Alfred Hitchcock Presents, The Waltons, Dallas, and Diff'rent Strokes. He also continued to do voice-overs for various television shows and cartoons.
He is currently a member of the Premiere Speakers Bureau and is promoting his own motivational book, Go Get 'Em Tiger, which has sold 30,000 copies via his internet website.