The earliest that you can unlock a 'Vendetta' mission is once you can free roam in chapter one.
Simply walk up to any enemy and allow yourself to be killed by them. This will allow the enemy to rank up to a captain which will unlock the Vendetta mission (Orange Icon) in which you will have to hunt down and kill the enemy that just killed you. Be warned that they will be upgraded with new weapons and be a much higher level then they were before so I would suggest being level 5 or higher unless you are good with your reaction commands.
Here is a video showing how it all works if you still have some trouble.