These count across all your playthroughs on any difficulty and it doesn't matter if you're in co-op or solo. You can check how many of these kills you have by going into My Page > Statistics > Campaign Statistics. They are listed under "Enemies Defeated (Bosses)". Unfortunately, there is no "total" for these stats, you will have to count each section across all difficulties to get a count.
The end of every episode has a boss, as well as the end of a few chapters. You can also kill the same boss over and over and each death will count as a boss kill. In order for this to count for you, you need to be the one who gives the final death blow. After the chapter ends, the player who gets the boss kill will have an image of the boss by their score.