Etymologically it's rather nasty: the word was coined in either Spanish or Portuguese as "mulato", as a reference to mules ("mula" - horse x donkey hybrid).
In Portuguese it seems to me that this association faded away. However, I'm going to take a guess and say that the word is probably a slur in English, so not something that you want to use for its meaning.
I'm a native pt speaker and I had never thought of the word as slur. I remember it being commonly said on TV, music, and written on newspapers without this connotation. It was certainly more common than the preferred alternative "mestiço".
Yeah, in Portuguese it is not a big deal. (I'm also a native speaker.) Some people do complain about it, but it's typically based on etymology, not current usage.
Dunno how others interpret it but for me "mestiço" isn't quite an alternative, it's more like a hyperonym for any person with mixed heritage (not just Afro+Euro).
I've only encountered the use of mulatto once in the wild. My girlfriend my freshman year at college had an adopted brother who was biracial. She used the word mulatto to describe him, but at the time everything she said about him seemed very loving.