You can go for this achievement by playing public games in online adventure or freeroam. An event is a race or a competitive game and you get progress towards the achievement when it finishes.
If you want this fast, it can be easily boosted by getting one other player, starting a private freeroam and playing a game of Infected. Timber Mill small is the smallest map so it won't take long for the infected car to catch the non infected one. Repeat this as many times as you need. It took me 45 minutes for all 25 games.