Same strategy as "Delivering Meat Bags" et al.
The best time to do this is once you have a ship capable of holding multiple passenger cabins while maintaining a decent jump rage, eg the AspX, Beluga, Anaconda; as this will increase the amount of missions completed per dock.
Make sure you're also allied with a decent station (have maximum reputation with a station of high system population - 1 million+ is a good amount) as these will usually offer more missions to you than stations you are neutral with.
As you are looking to get a higher number of missions done rather than a higher payout, you'll need to skip VIP/Explorer passenger missions as these are not only very time consuming per mission, but 9 times out of 10 you will have to do them one at a time - they won't go to the same place. The meat of this one will therefore be short range economy & business transport missions: Missions such as "X politicians/refugees/tourists/etc seeking transport". Stack up as many as you can fit into your hold, and prioritise those that not only go to the same system, but to the same station.
This achievement will of course be very time consuming and repetitive. Have fun!