This is pretty straight-forward:
You can access the "Daily Run" via the menu. Select the current day and pick a class/build. After that, you are prompted to select a map. Any choice will do here.
When playing, you don't actually have to complete the whole run. You can fail at the very first stage without getting any real score. The achievement counter will still consider the failed run.
The achievement should pop right away after your first daily run.
Unfortunately, there is no integrated progress tracking available as of the creation of this guide. You can check your progress though by navigating through your daily runs in the menu. Keep going for a total of 7 days to get the achievement.