Pretty much as the achievement states, get the final winning kill in arms race. For those that don't know, arms race is a mode that has your team cycling through a series of weapons having to obtain three kills while armed with said weapon, because of potential update changes I won't go into the specific weapon order but for the achievement, while armed with the final weapon in the sequence you have to get the final kill that wins the round.
Fortunately this is relatively easy to set up as this achievement is obtainable in private matches, grab 4 friends to play with if you can otherwise fill the game with bots but increase the number of rounds to win if you wish just in case a bot on your team keeps pulling off that final kill.
Once in the match, fight it out and just try your best to pick off the final kill in the round to pop the achievement.