You need to have Poison Weapon active in order to right click and add poison to a weapon, but AFAIK you do not need to keep it as your secondary skill.
I did not use this skill, so am not sure if the game applies the poison to the specific weapon you have equipped or treats it as a bonus to any equipped weapon until it runs out. To test this, simply poison a weapon, try it once, then switch weapons and see if the poison effect continues.
