An easy way to get this achievement is to play a griffball match, and get to the back of the field (the enemy base). You can kill them all as they spawn, and if you can keep it up while the enemies at your end of the field get wiped out, you should be able to get extermination.
This is also an easy way to "killtacular" as you can kill 5 enemies easily.
Another way to get extermination on griffball is to defend the bomb carrier as he drops the ball into the enemy base. The bomb blast has a habit of wiping out the enemy team, and if you are lucky enough to put some splash damage on the enemies with a hammer the game thinks that you were the one to kill them all. If you go in with a double kill, you can sometimes get an extermination this way.