Well, for printing you shouldn't have to remove the background. Most printers don't print white.
But if you want to remove the white I'd use Select > Color Range. Make sure to check contiguous so it doesn't select the whites of the eyes (unless you want that removed too). If it's a perfect white you can move the fuzziness all the way down. Otherwise slide it up to get the detail you want. Delete it, then save as a format that preserves transparency (I believe GIF should work?)