Time Trials are an almost hidden area of this game. In order to access them you need to go to the leaderboards from the home menu and select time trial. Then select the leaderboard/race that you want. Pressing 'A' again will start you in the race for that leaderboard.
Simple enough but a little hard to see at first. You need to complete at least one whole lap, without using the rewind feature, for the achievement to unlock.
It should also be noted that you will need to be online in order to see the leaderboards, but as I virtually never play offline I wasn't aware of this.