I'm not counting Ricky Nelson here. He was a good singer and could even play the guitar. His TV series made him a hit, but I think he'd have made it without TV.
A little later on, Richard Chamberlain played Dr. Kildare and became a recording star with "Three Stars Will Shine Tonight." And then there was David Soul, who was either Starsky or Hutch. I forget which one, but I'll never forget "Don't Give up on Us." And then there's John Travolta, who was a better dancer than a singer but who did have a hit or two, like the classic "Let Her In."
John Schneider was no Elvis, but he recorded "It's Now or Never" and a lot of other stuff, too. Not a bad singer, but would he have made it without The Dukes of Hazzard? How about Bruce Willis? Without Moonlighting would he ever have recorded "Save the Last Dance for Me"? Tom Wopat, Schneider's co-star on Dukes had a hit with "A Little Bit Closer."
I'm sure you can think of a lot of others. Like William Shatner. But he doesn't count. He didn't ever have a hit. At least I hope he didn't.
Update: How could I have forgotten Peter Fonda? And thanks for all the names in the comments. I can't believe I left some of those off the list.