To get this achievement you need to beat the "twins" in their one-on-one just like the achievement says. This can be done on any difficulty and is obviously easiest on easy. To do this you first need to unlock their races by getting 185 lights, you get lights by winning races and doing fan runs. Second you need to complete their career demands which are, hit 5 cars with forward shot mines, nitro ram 5 cars, get 10 bonus hits in destruction, and wreck 1 car and win in event 6. The forward shot mines need to be a direct impact on another car to count, so aim at a car and hold forward and fire. Pick up nitro, wait until you are behind a car, and then nitro. In destruction you get bonus points for doing combos, so if you shoot a yellow truck and it drops a mine which kills a green or yellow car then you get bonus points, you can also drift wreck cars to get bonus points. The hardest one is wrecking a car and winning, do this on easy and put the mod on for increasing shunt power, collect 3 shunts and go behind a car, make suret they dont have a powerup to block the shunts, then just fire all 3 into them and they will most likely wreck unless they are in a truck.
You have to use the nissan skyline GTR for this race, which isn't a bad thing. Now this is a 2 on 1 race which makes it harder than the other one on one's. One thing to remember though is the twins will be able to hit each other with power ups, I found once i got into first they kept each other busy and didn't bother me and i won with a ton of space between us. To get into first try to pick up 6 bolts and a shield. Bolt them both, then shield and try to take them on a corner just to get infront. Then try to pick up more bolts and keep shooting them behind you to try and slow them down. After your in the lead its a pretty easy race because they will attack eachother. Just cross the finish in first and this is yours. And remember if you cannot beat them to the finish recking the rivals car (both cars) will give you an automatic win, so if you can, wreck away. If you plan to wreck your rival the easiest way is to turn the mod on that gives you 4 bolts, gather up 3 bolts and fire all 12 into one of the twins, this should wreck them, do this again to the other to wreck both.