You can earn this trophy by going through the story mode, or if you want to get this trophy without completing the game use this guide.
To bag yourself a revenge mission you need to piss someone off.
For example, a person has kidnapped one of your operatives because you hospitalised them and they want revenge.
The easiest way to do this is to have a nice list of operatives, then you need to get people to hate you, that it's represented by two red thumbs down icons when you profile them.
You can get them to hate you by running them over, knocking them out ect.
You need to wait a while, as every npc has a life of their own to live. So it may take a few days before they decide to act.