There's not an automated, round based, tournament mode for Code Rally. When we looked at automating it we realised that the random factors (starting positions, opponents in each race) means that only looking at one race to determine a winner isn't a good/fair solution.
When we have been running online contests over the last few years we have looked at the fastest race time for a specific track, and use that as a way to determine who has the best AI - this allows people to run as many races as they like, so that random factors don't ruin the chances of a good AI. It also encourages people who have gotten a good time to experiment with new solutions, as their best time won't get any worse and they may be able to better it by doing something different.
To run a contest based on fastest time you simply just need to have everyone race on the same track and on the same server. On the leaderboard look at the track you want people to race on by selecting it in the track dropdown - this lets you see who has the best race time for that track.