Halo 3

The Legendary Maps are Ghost Town, Blackout, and Avalanche. The goal here is to get 5 kills with the Warthog's chaingun in one match.

The achievement may as well have said, "Get this achievement on Avalanche," because as far as I know there isn't Warthogs on Ghost Town (and there aren't any Warthogs on Blackout). Pretty much, keep playing the DLC playlists until Avalanche pops up.
Now, hop on the turret for the Warthog. If it's a team game, hopefully someone will drive you around (and hopefully they're good at doing so). If it's a free-for-all, then you're a sitting duck.
Note, though, that these 5 kills don't have to be in a row. So if you get killed, feel free to continue with your chaingun massacre. Another note: if by the powers that be a Warthog with a Gauss cannon is on the map, don't expect the kills from it to count. The achievement specifically says "chaingun."

This is a pretty easy achievement to get. Hopefully you won't have to go through the motions twice, like I had to. I got 5 chaingun kills once before and I never got the achievement. Worked the second time, though.

YOU CAN GET THIS ACHIEVEMENT ON THE MYTHIC MAPS AS WELL AS ALL THE OTHER DLC ACHIEVEMENTS!!

All u have to do is get any type of game on sandbox, but make sure its a team game. Immediatly get in a warthog and have someone drive u around while your in the turret and have them drive around the map while u pwn on the unsusspecting spartans =] 5 turret kills then u have this achievie =] good luck

Actually, I got mine on Sandbox (don't ask me how). It doesn't matter if its in a row or not. Just kill people (while as a Warthog gunner). You can also be stationary (not moving). Just be sure you get the 5 kills you need.

This can only be done on Avalanche.
The easiest way to do this would be to go into Social Big Team. This improves the chances of you getting Avalanche as it is one of the larger maps and if the game is Slayer gives you more time and more targets to get it with (100 kills rather than 50). If you get really lucky and its a CTF game, then you could go nuts killing people if you decide to go for the achievement first and the games objective second, thought thats not really in the spirit of the game. Just get a friend, or if your playing alone with 8 on a team somebody always ends up driving, hop onto the Warthog, and i suggest staying close to the center passage between the two sides of the map, as people generally rush through there. You should be able to get this easily the first time you try. Remember this does not have to be a spree, so if you die get another Warthog and try again.

Ctf is the easiest playlist to attain this on as players are focused on the primary objective... not that you should be ^-^. anyway this achievement is applicable to any dlc released after the legendary map pack as well. if the warthog is positioned well in ctf the achievement is easy due to the enemy coming to you making them prime targets. if you die its no problem just wait for the warthog to respawn and continue the slaughterin.

Because one of the map-packs was released earlier, you can get Legendary achievements on Mythic maps, but not vice versa. Just play Social Big Team, and wait for either Sandbox or Avalanche to pop up. Hop on a Warthog, and start popping players.

If there's no one to drive you, get yourself to a safe place where you can see danger come from far away.

This, and all DLC achievements can be done in the Team Mythic playlist (except the Delicious Brains and Zombie Repeller, which must be done in the Infection playlist, Tank Dropper and collecting the skulls, which must be done in Forge mode, or Killtacular which requires 5 opponents, either in a free-for-all or Big Team playlist). All DLC achievements will unlock on maps from later map packs.

Team Mythic requires 6 players, and maxes out at 8 players. You'll have to get 2 players together with 4 controllers each, 8 players, or somewhere in between. If you can't get 8 players and/or guests, you can do this with only 6; however, you may need to put up with randoms, but they will likely be willing to help. After all, who would still be playing this mode unless they were achievement hunting.

For other details how to get this achievement, I'd refer to the other solutions here, but the only map that comes up in this playlist that has Warthogs is Sandtrap.