Your best bet with this one is to find another racer ASAP after the Eliminator starts.
If you are lucky enough to spawn near a car drop straight away, then grab it before you challenge someone - it'll only improve your chances quicker!
Everyone should start with a red Mini. Race another player by honking your horn at them (pushing down on the R3 stick), and you'll obviously need to beat them.
TIP: All race finish lines should be towards the middle of the battle royale style shrinking circle arena so make sure you are facing in that general direction if you can before you challenge.
Once you have beaten them, you will have the chance to take their car (obviously don't bother if they only have a Mini) or upgrade yours to the next level - so a Mini is level 1, and you'll be given the option of a level2 car. I don't think I need to tell you what choice to make there!
Then, start chasing down and challenging other players, just be careful not to challenge someone who has managed to pick up a level 6 or 7 car drop already as they are out there.
That said, I got challenged by a level 7 when I was in a level 3 car and I won the race as they hit a tree right by the finish line, giving me a lucky win (guess what car I chose to take next at the end?)
Each win gives you the option to step up to the next level or take the loser's car. That choice is up to you.
If you're lucky, you'll beat 3 Mini drivers early on, or other low level cars before they manage to upgrade.
If someone is driving away from you, they probably don't fancy it, so chase them down and grab those wins!
Also be aware that, dependent on the season, some vehicles are better on the icy lake for grip, but don't have great acceleration on the roads.
All you need to do is win 3 races in one session and the achievement is yours. Just go again in another Eliminator session if you don't get it quickly - it took me about 30 sessions, but I was also searching for car drops to try and get that achievement too.