Eddie's Challenge consists of 15 events. They consist of circuit races, sprint races, time attacks, and drift contests.
First, you will get a call from Travis telling you that Eddie has set up a series of challenges. Then you will get a call from Melissa and get the Challenge App (accessible from the start menu). Each time you complete a challenge, access the app, and it will give you the location of the next challenge on your map.
You need to come in first in each event to move on to the next. Thanks to patches to the AI, it isn't as bad as when the game first came out. You should easily beat most of the events with your fastest car. For the drift contests, Ken's car is useful if you don't have a good drift car/build of your own.
When you finish the final race, the achievement will pop once you load back into free roam.