Image: UbisoftOfficial screenshot from Ubisoft.
Toxic Influence asks you to have poisoned soldiers kill 5 other enemies. Poison turns FND troops against their own side, so this is about setting up fights, not shooting.
Poison a soldier who is standing near other enemies. Let them turn and fight before you open fire. Every kill a poisoned enemy makes counts toward the 5.
Pick crowded spots like checkpoints and bases where soldiers bunch up. Throw poison into the middle of a group from cover, then stay hidden and let them fight. Shooting too early can kill your poisoned helper before he does the job. Poison ammo for compatible weapons works too.