Boris Karloff did not do the singing
Karloff is credited with all the speaking parts, including the singing, but I saw a special about the making of the Grinch, and another person did the singing of "You're a Mean One, Mr. Grinch". They had him talking about it on this special.
I don't remember his name, but for those old enough to remember, this voice-over actor is the same one who played Tony the Tiger in the old Frosted Flakes commercials. Remember the way he said, "They're GREAT!" When I hear the singing on the Grinch, I can tell it is him. Deep rich voice.
For some reason he was not credited and the credits make it sound like Karloff did the singing. He said Karloff apologized to him.
You must be the change you seek in the world. -- Gandhi