Henry is Barney's 1/2 brother. Same mom different dad's. His dad was a monster of a dragon, but that's a whole other story. You know how they say "What happens in Vegas, stays in Vegas"? Well Henry happened while his mother was out there for a publicity stunt with his older brother Barney.
So with a mother who loved him, but was constantly reminding him that he was an "oops" and a famous brother who WOULD NOT STOP SINGING THAT SONG—you know what one I'm talking about—Henry had some teen angst years where he would set things on fire and try to blame it on his brother.
But after a near death incident when he sneezed at the gas station and was saved by none other than Mr. Purple Dinosaur, Henry turned himself around. He got a whole new outlook on life where he realized he could enjoy his friends, family and life in general way more if he took that never ending song to heart. And he finally said "I Love You" back to Barney.