In Party Games I recommend One In The Chamber. Each game including map selection etc. in the lobbies is about four minutes so you win or lose quickly. For the Achievement you just need to finish in the top three, ten times. Often, though not always, four kills is enough to do this. If you're good enough to place in the top three consistently the following tips aren't for you.
At the start of the match check the mini-map to see the closest enemy. If you can win that one on one you may only need three more kills with your three lives. Find a spot with just two ways in and out, and watch your mini-map to see if anyone is coming. Keep moving around that area so you don't appear as a static target. Where possible, use your knife for more points. If you see two red dots nearby, try to get a kill there, since you know they're pre-occupied with each other.
Playing like this you'll be collecting Survival bonus points also. This can be important because in one game I showed fourth on the scoreboard during the match but at the end I was listed as third, I guess because I had more Survival bonuses than the guy who was ahead of me with more kills.
You can check your progress via Barracks -> Combat Record -> Game Modes.
Previously I tried Gun Game and had no joy there. One In The Chamber is definitely easier to land near the top, and it's also faster.