Buddy-Up attacks are a new part of Lego The Hobbit game. A buddy-up attack is having two compatible characters, such as two different dwarven characters, selecting one of them, then walking near another one and pressing . You should then see yourselves connected to the other character standing side-by-side. Then press the to use the Buddy-Up attack and kill the enemy. You have to kill them for the "Buddy-up attack" to count.
Do this 50 times over the course of the game or you can farm this on Chapter 3, "Azog the Defiler" if needed.