Go to the "ACTIVITY FINDER" and then select "BATTLEGROUNDS".
Depending on the time of day an your server region, you may find matches quicker or slower.
Chaosball is a new mode added to the Battlegrounds rotation and during the time of this challenge I mostly got that as the game type. Deathmatch was the second most common with the other two never popping up.
You can quit out if you don't get Chaosball but you'll be banned from requeing for twenty minutes.
What you have to do in the mode is pretty simple, run to the center of the map and grab the "ball" once it spawns and then survive to build up points till your team reaches max points or the match timer runs out and whichever team had the most points is declared the winner.
Typically matches shouldn't run the whole time and a team should max out.
The only catch for this challenge is that you need to be the one holding the ball to gain progress. It doesn't matter if you win or lose, just hold the ball for a good amount of time, I personally racked up about 40% per match.
What the above means that even though it's counter-productive to how the mode is supposed to be played if your teammates ever have the ball then let the enemies kill them. The more points someone on your team gets is less potential points that you'll get.
All in all this shouldn't take you any more than about two to five matches depending on things like luck and skill.
That said I still wouldn't hold off on getting this done, I've had problems just signing on to the game and the queue times are unpredictable.