In the main menu, go to "Online Adventure" and then select "Ranked Adventure." On the right is an option for "Ranked Free-For-All Racing". Click on it to begin. Complete 10 of these (5 rounds in each one for a total of 50 races) to receive your rank. There is a counter at the bottom letting you know how many of the 10 qualifying adventures you have completed. Sorry for the confusion in my previous solution.
+ You will not have access to "Ranked Adventure" unless you hit level 3 in "Racing Online Adventure." So to level this up, you'll need to complete a few unranked ones. To do this, in the main menu, go over to "Online Adventure," select "Quickplay Adventure." and then "Free-for-all Racing." Thanks to Oriole2682 in the comments for this.