In order to attempt this achievement there is two things you can do.
1. Progress a large amount of the story to unlock the 10 bosses yourself and simply defeat them.
2. Join a random lobby and hopefully they are doing all the bosses.
Somethings to note:...
is that you can set your own lobby to private and do these by yourself providing you have the bosses unlocked, as the game gives you AI's to fill the lobby.
These don't have to be done in order, if you join a random online game and beat a boss but then play solo you don't have to repeat the boss. So feel free to mix and match.
In the Governors Office/lobby on the right is the Co-Op kiosk.
The bosses are, so you know once you reached your 10.
Recommended Level: 25 Angry Berserker
Recommended Level: 30 Judgment System
Recommended Level: 35 Shooting Phage
Recommended Level: 40 Piercing Eyes
Recommended Level: 45 Ruthless Angel
Recommended Level: 50 Dirty Scissor
Recommended Level: 55 Mighty Chariot
Recommended Level: 60 Ghospet
Recommended Level: 65 Subjugator
Recommended Level: 70 Corpse Sniffer