He has been mentioned several times throughout the series. He has been implied to be abusive, and frequently beats Joseph's Mom. He has also been implied to be an alcoholic, as he spends all the Christmas money on booze in The Christmas Special!, and kicks Joseph out of the house.
In The Alien!, it is revealed that he and his wife died of unknown circumstances, leaving Joseph orphaned.
In Joseph's House!, it is revealed that his dad used to be a Koopa, and is now a ghostly Dry Bones. He can now turn in to a boo.
In Joseph's Mom! it was revealed that Joseph's mom didn't actually die, meaning that Joseph's mom is actually alive and she didn't die from unknown circumstances, and the skeleton that lives in Joseph's house is actually a Halloween decoration.