It just means that he has a bad character. The simplest synonym is bad. There are many literary synonyms and idioms that express this idea. It is nothing to do with bad mood which is temporary.

I suppose it depends how that bad nature manifests itself. Bad tempered is definitely more common but it means quick to anger, whereas bad natured, is more general and can express any kind of irritability or unpleasant character trait. Irritable Disagreeable Grumpy Prickly Moody Rude Annoying Are all words that could fit in UK English I would be more likely to use a noun though. "Maybe I'm trouble", "Maybe I'm a pain in the neck", "Maybe I'm a bit of an arsehole", "Maybe I'm a nuisance"

"To be born wicked" is a bit too strong, I think.

Google "Synonym for bad natured" and you'll get lots of suggestions. "Bad natured" would be more likely to refer to a group of behaviours that might make up a person with an unpleasant personality. If it was a specific unpleaseant thing, like meanness, dishonesty, crudity etc it would probably have been stated as such. Hmmmm... actually "nasty" is a pretty good synonym for "bad natured" as it could encompass more than one unpleasant characteristic.

To mean, this means that the singer is mean, a person with inclinations towards bad behavior. A ´´bad-natured´´ person is someone who inclines towards evil...liar, thief, always in a bad mood, etc.