Before attempting this, you will need to research/build medikits and equip at least five when on the load out screen. It is highly recommended you use a Support soldier with the 'Field Medic' perk as this will alow a single medikit to be used three times instead of once
The easiest way a soldier can become poisoned is by a Thin Man so it is a good idea to attempt this in an early objective based mission. (The Thin Men seem to be the main enemy in these missions.) There are two ways to be poisoned, either have a Thin Man spit poison at you or after you have killed one, move your soldier into the poison cloud it leaves behind.
Once a soldier becomes poisoned it must be treated by a soldier carrying a medikit. Just be sure that the soldier running into the cloud can't be carrying a medikit as this will make them immune. When five soldiers have been cured of poison, the achievement will unlock.
TIP: An easier method to achieve this is to try and isolate a Thin Man and have at least five of your soldiers close by. Have your first soldier move in close and make the kill. Once the Thin Man is dead and has left behind a cloud of poison, move each member within the cloud. On your next turn, set about curing everyone. Be sure to make a save before attempting this as it may take several tries.
Note: This does not have to be 5 soldiers. It can be done with 1 soldier, 5 times. Just do a Bomb Disposal mission with all Thinmen. Had 1 soldier run in to the poison and out, cured then back in to get poisoned again. Done this way, each cloud is good for 2 poisonings.