I think Jerry Van Dyke, brother of Dick Van Dyke, who is also a comic actor (he was the sidekick in the TV series "Coach"), plays guitar and sings and had a couple of hit novelty records back in the 60s.

Charlie Chaplin composed music to accompany his silent movies, and you can buy albums of his music – called by the slight misnomer "soundtrack albums".

Steve Allen played piano and composed lots of music.

Robert Duvall did his own singing in, and some songwriting for, "Tender Mercies."

Slightly off-topic: Dwight Yoakam is a decent actor and did a good job in "Sling Blade." (I see he has made several other movies, but I haven't seen them.) Willie Nelson was credible, though not very exciting, in "Honeysuckle Rose" and "Barbarosa." Paul Simon was quite good, I think, in "One Trick Pony." (Though the soundtrack album was a dud.) Courtney Love was excellent in "The People vs. Larry Flynt."

Neither here nor there: Harry Blackmun played a Supreme Court Justice in Amistad, and was suitably dignified. The wonder, of course, is that he did it at all.