Given how glitchy the achievement tracking is on the public playlist I thought I'd try an easier alternative. I only had 1/5 despite having 7 wins logged in my tour of duty but I was able to get the remaining 4 I needed in custom games.
Select the following from the main menu
VERSUS --> CUSTOM --> LAN GAME --> HOST GAME
on the GAME SETUP box in the lobby to change the game optionsGame Setup
Game Mode: ARCADE DEATHMATCHArcade Deathmatch Options
Fill with Bots: ON
Bot Difficulty: BEGINNER
Rounds to Win: 1
Round Score: 10
Execution Rules: OFF
Respawn Time: 10
DBNO Time: 15
Friendly Fire: OFF
to go back to the lobby then
to start the match. You should see progress on the achievement tracker after each match.
Note it looks like you need to win 5 matches while in the custom game lobby, which might explain why some people aren't seeing the achievement tracker update. E.g. if your achievement is already at 40% (2/5) you will still need to win 5 matches before leaving the lobby.
Thanks to Xerostomia and Fed kaffe for clarifying this in the comments.
Note you can lock all the slots on the opposition team, which gives you an instant win since there are no enemies to kill when the match starts.
Thanks to Jayhelm for providing this tip in the comments.