You can get this in single or multiplayer. I will give suggestions for how to quickly get these for multiplayer since there is so much to do there for achievements already...
First off the fastest way is going to be to run with a group of 4 players, specifically people you know or set up a session with who have mics and are focused and will rush for mission objectives.
If you get the mission to clear out X location. Have one player run into the room and aggro everything and pull it out of the objective area. If the zombies are outside of the area then they do not count toward the stuff you need to kill in order to clear out the objective. Then have 1 or 2 people run in and kill whatever zombies are left over. Chances are after you clear out the objective you are going to get a new mission to kill X amount of zombies. Now you have a big pile of them just outside the door where you pulled the mob to in order to kill quickly for more points.
Be sure to buy the PP upgrade in the skills tab when you are able to. Sadly they are level locked so you can't buy them all right away. There is a multiplayer trial for Day Tripper which is to complete X days successfully which means escaping to the safe house at the end of the day. The first 3 ranks will unlock PP bonus skills which raise the amount of PP you get in multiplayer. It takes 25 completed days to get all 3 levels.
The last thing that will help you is making sure to get to the exit! You get 1000 PP times the number of people who make it to the exit in your game. So if there are 4 players then you get 4k exp every time you finish a day. This is a large boost in exp!