Halo 3

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.

The second easiest but more common way to get this is in SWAT. headshots are instant kill, in SWAT you can be a one man army. just flank the enemy (go the long way and get them from behind or to the side) as they wouldn't know it and also in SWAT theirs no radar so they can't see you. or you could just go in like a headless chicken and try to blast them front on (it's how i got mine lol).

than the easiest way is in griffball but it can only be done on sanbox for this achievment (foundry isn't mythic) just spawn kill them.

also the ENTIRE opposing team must be dead (respawning) to get this.

if it's like 3v3 you can still get it just get an overkill on them and you should get it.

Okay the two gametypes easiest to get this in is Griffball and SWAT. I have seen many exterms from both of these game types from players.

First for Griffball when you play it you'll know its easy to get loads of medals as long as you get the right timing between swinging your hammer and the correct distance between you and an opposing player. Your best option is going into the playlist: Action Sack. You get 0xp in this playlist but if you want the achievment then it's a good idea to go in it. When actually playing Griffball aim to stay at the opposite side of where you spawn, the other team's base. This way you'll do what's called spawn killing. Spawn killing the other team is when they have just respawned so they don't have enough time to do anything before you hammer them to shreds! Hopefully if their team doesn't have the bomb and your spawn killing you'll get the extermination and probably along with the killtacular medal. Two achievments in one game. Infact three, if you get a bull true on a mythic map.

Now secondly game type SWAT is easy for this. If you have quick reaction times an exterm can be picked up. In SWAT you have no sheilds and no radars. This means many exterms in SWAT are gained by coming behind the enemys. All it takes is a headshot in SWAT for a death. Put it this way: Pretty much every match at least someone gets a triple kill, multi kills come easy.

Okay so remember that an extermination is obtained by killing all the members of an opposing team with at least an overkill and they all have to be dead at the same time (without any of them having respawned) Exterminations can be pure luck.

I got it by playing a Grifball. However, by now, it won't be going on anymore because they will end the double EXP weekend on September 14th, the release date of Halo Reach. I may be wrong though because Bungie might add it in to the social playlist. However, if it does end and never come back after the 14th, do this.
Go into a social playlist game that doesn't include very many people. The place you want is Sandbox. If you can get a game on that, then you can go to the other base on the other end and just wait for them to come back and respawn like Grifball. They might not respawn there though so it may take some time. If you have any friends or teammates over there, that would help as they rekill the people on one end and you rekill the people at a different end. Eventually, you may get an Overkill or Killtacular and thanks to your teammate, you may have both knocked out the other team people and you may have an Overkill or Killtacular to show for it. Since they are still respawning, you will get an Extermination. This is a very hard way without Grifball, so sorry if you can't get it.

this is isn't the easiest way but you should really get the acheievmwnt on grifball

but i got mine on a flag match on orbital if your attacking you should have a ghost on it use that and charge the base where everyone respawns

I got this achievement On Longshore Mythic Team deathmatch (slayer) 5 VS 5 it was I believe. Just grab the rocket launcher. Wait for a group of them near each other (its all about the timing and just skill/luck)