First of all, the objective of this achievement is to win a match in each of the following game types;
Domination - Capture areas and gain points, first to the score limit wins the match.
Team Deathmatch - Split in to two teams and each have to battle to kill the opposite team, first team to reach the kill limit wins.
Free For All - Basically this is the same concept as team deathmatch apart from the fact that everybody has to kill each other.
Ok, now that's cleared up. There are two ways to obtain this achievement. You can either just get this randomly whilst playing with friends online or on your lonesome. Or you can take yourself in to private matches and work to get this achievement. Now if you are not patient then I would recommend doing this in private versus, which is what I did.
Ok so to do this in Private, these are the fastest match options I could come up with when I tried to get this achievement.
Head to versus - Private, choose Domination;
-Round score limit: 50
-Number of bots: 1
-Bot difficulty: Casual
-Friendly Fire: Off
(Choose any map, when the match start capture all three areas, and just wait until the match ends. The bot may contest the areas so just make sure you kill him and re-capture)
Head to versus - Private, choose Free For All;
-Goal Score: 10
-Number of Bots: Fill
-Bot Difficulty: Casual
(Choose any map, when the match starts then just go crazy and kill everybody, you should be able to rack up more kills than the bots)
Head to versus - Private, choose Team Deathmatch;
-Goal Score: 10
-Number of Bots: Fill
-Bot Difficulty: Casual
-Friendly Fire: Off
(Choose any map, when the match starts then just kill the opposing team, your team will help but it is faster if you just find and kill the enemies)
Ok, so just use these match settings, complete the matches, or it will not work. Once you have done this once for all three then the achievement will pop.
(Edit: Thanks to @MuddyAmoeba for pointing out that the party type must be on either; Joinable friends, Joinable public or invite only. And cannot be done in split screen or local mode. To change this press the back button in the menus)
Thank you for reading the solution, I understand that some people will not need a solution for this so please do not vote negatively, this is in place for the people that are new to gears and need a little help. Positive feedback will be appreciated. If you leave a negative vote then please explain why.