For this achievement, you have to deliver 7 or more newspapers in one day, Hard Way mode. That means you only get to miss 3 or less houses. Do this on Monday (in-game) where there would be less obstacles than the other days.
I recommend to not go too fast and not too slow, just something split down in the middle. Going too fast will make you even more vulnerable to obstacles (unless you are crossing the road in the middle of Hard Way and trying to avoid the maniacs that try to tackle you down before you cross). Plus, you can miss many houses. Going too slow causes the swarm of bees to appear and attack you. A delivered newspaper counts when you throw it at the subscribers' doormats or their mailboxes. What doesn't count as a delivered newspaper when you trample their plants, break a window, throwing the newspapers the wrong way, and ignoring them.
Practice the Hard Way - Week 1 - Monday level. You start to learn patterns of obstacles and what to do to avoid them, just like I did while trying to accomplish this achievement. The obstacles never randomize in different game sessions. Pick up bundles just in case you start trying multiple attempts to deliver a newspaper to a subscriber's house.