Across all of Superman canon (comics, movies, TV series), explanations for both Superman's superhuman abilities and his vulnerability to kryptonite have been made multiple times. Are they always in agreement (i.e. our Sun's radiation) or have different writers gone different ways when it came to explaining the source of his strength, or the reason kryptonite robs him of his strength?
Short answer - no.
Longer answer - over the years, various creators have attempted to come up with more plausible reasons for Superman's powers. They have varied wildly in success.
In the Golden Age, at least one version of Superman's origin said that Kryptonians had at least some super-powers on Krypton. The Science Council scoffs at Jor-El's suggestion of migration to earth, citing that Earthlings are so simple and unadvanced, "They do not even possess X-Ray Vision!"
John Byrne posited the idea that many of Superman's powers were psionic in nature. He doesn't fly, he levitates; his strength has a facet of telekinesis to it, etc. During his run, Superman's invulnerability was due to a naturally generated force field that extended a fraction of an inch from his body. This field protected his suit (at the time it was not made of Kryptonian cloth) but not his cape, resulting in his capes being shredded all the time (and a truly hilarious scene where Ma Kent had made up a new batch of capes for her son, handing him a yard-high stack of red cloth.)
The most common thread between all the versions is a combination of Earth's lighter gravity and increased solar radiation combine to give Kryptonians (and Daxamites) their various powers. The gravity is generally seen as source of the strength and durability, and the solar radiation as the source of the more energy-based powers like heat vision.