Note! This quest is optional, but its completion can bring you closer to unlocking next major Companions quest.
Possible quest givers are Vilkas or Skjor. After you agree to complete the task, you'll be asked to kill an escaped criminal in the random location of the game world. Leave Whiterun and go with fast travel to the location which is closest to the criminal. Some difficulty of this quest is a fact that the bandit is in constant move, so all your decision must be done with a certain advance. You can easier catch a bandit on a horseback (screen above).
Depending on your preferences, you can attack bandit without warning (screen above) or after a short conversation with him. You shouldn't have any troubles with defeating him. You can now return to quest giver and collect your reward in gold.