At first he wanted to get strong to help people in need, but it was really on a whim. He effectively decided to start his superhero regiment for "fun". Then he became disenchanted by how immensely strong he'd become, his hair fell out, the public got kinda weirded out about him, and that new isolation was something he couldn't tolerate.
You may recall a scene where he was extremely bored trying to save a guy who was about to jump off a roof. When the dominance came, and the hair fell, that "fun" went away. He had achieved the pinnacle of his hobby, but he was painfully alone.
But then he encountered Supersonic, who shined light on Saitama's lack of direction, creed, and identity. Sonic challenged Saitama to get serious and do something with that cosmic strength. So, on another whim, he joined the Hero Association, hoping he might find a way to curtail depression (and to make some extra scratch for bargain bin sales).
His real motives are often muddied by comedy. "YOU GIVE BALD GUYS A BAD NAME!" "Those.... were MY FRIES!" etc.
We always look forward to his slapstick. In this chapter's cut-away mine kart scene, Flashy Flash (who still hasn't seen Saitama actually fight) misses the mark when he assumes Saitama is after fame. It's pretty clear he just doesn't want to be alone. He wants friends, the recognition of his peers, and the respect any lower-middle class Japanese salaryman on the brink of being swallowed into otaku culture. Saitama became a hero to not be alone anymore.