This can be a difficult achievement as finding a 'very hard' abduction mission can take time. To increase your chances, try and raise the panic level in one continent. The higher the panic level, the harder the mission will be.
As there is a limited number of maps and the rotation is random, it may also help if you wait until you receive a map you know well and are confident you can get through quickly. Just be sure to save your game at the base before launching the mission. Your soldier load out is entirely up to you but it is highly recommended you have a minimum of 5 soldiers and trying to clear a very hard mission with only 4 is near impossible.
The easiest way to beat the mission in five turns or less, is to start off by proceeding through the map learning what enemies are present and where they are located. Then reload the save you made at the start of the level and run through with that knowledge. Their positions and strategies may vary slightly but overall you should know what to expect.
TIP: Despite the option of Classic or Impossible, this will be much simpler on Classic