When you finish a Rally (all 4 stages) you can replay the stages and press to bring up the rivals mode. From there you will be given an opponent who has driven the race faster than you. You can make it even easier for yourself by changing the opponent and changing the filter of the leaderboards to everyone and choose the first guy above you. Shouldn't be too hard if you can beat your own time.
Every stage counts for this achievement, you don't need to win the whole rally for it to count. Also note that you need to beat your rival. The achievement description is misleading when it says "complete". If you lose, it doesn't count.