To "do someone wrong" is to do something bad to him, whether it is illegal (like robbing him) or simply unkind (like buying the last of something you know he needs). Literally it means "to do wrong to someone," and is very subjective. This one is often used in breakup songs (I've had "I Will Survive" in my head since I read the question).
To "do someone right" is to do something good to or for him, from simple kindness (saving his favorite food for him) to major assistance (going out of your way to prove his innocence so he can get the fire insurance to help rebuild his house). Literally, "to do right (or good) to someone."
Both of the above phrases are often used with the wrong verb forms, so it will probably be confusing.
To "put someone in the wrong" seems more complicated, and I'm not entirely sure about it. If someone did something bad and/or illegal, he would be in the wrong. If someone did something and you just learned that it was illegal, you might say, "That puts him in the wrong," indicating that the new information changed how you see him.
I don't know if there are any differences in British English.