This achievement can be done solo, but it requires a lot of luck with goat spawns. With 2 or more players, it's really just a matter of getting lucky on which spot the truck spawns in. Any difficulty above normal will have enough goats to get 6, but higher difficulties make it easier to find 6 goats without too much running around. Personally, I did this on Overkill with one other person. Any build should be fine as long as you can survive the difficulty you choose.
When the mission starts, move the car next to you forward to the park and open the trunk. Start looking around for the 6 goats you need. The ones you're looking for are any that are just sitting around. Some spots they can be found are in the park, on the sidewalks all around the park, and in the alley behind the convenience store. If you're on Overkill, there should easily be 6 that don't require doing anything other than picking them up. Load 4 of them into the trunk of the car and pick one of the spots where the truck can spawn (I liked the spot just down the street from the convenience store as it was closer to the park/goats). Park at that spot, grab the two remaining goats you couldn't fit in the car and throw them in that area too. About 3 minutes into the mission, the truck will show up. If it's where you parked the car, great! Throw all the goats in. If not, restart and try again. It shouldn't take more than a few tries to get the truck to spawn where you parked the car.
The achievement unlocks after the sixth goat is in the cage. You don't need to finish the mission.