Like it says, for this achievement you must deliver at least 7 papers in one day on Hard Way. It's advised to keep to default settings as this affects other achievements and prevents them from unlocking so it's best to be safe.
You really want to go for this on the first day (Monday) as that is when there are the most number of houses available to deliver to and the least number of obstacles. There are 10 deliveries available on Monday so just take it slow. If you try it on Tuesday (and beyond) and missed some deliveries on the previous day, those houses become non-subscribers so you have less opportunities to attempt to deliver papers. You can deliver either to the mailbox or the mat infront of the door.
It should be noted that this achievement is buggy for some people and won't unlock for delivering 7 newspapers. I had to deliver 9 to get it to unlock for example. So if it doesn't unlock at 7 try again to get more. Good luck!