To create the Poison, you need to collect alchemy herbs with the help of Herbalism, you'll need two of these herbs and one bottle of alcohol (like Cider or Beer). Now you can make drinkable cocktails with the help of Herbalism tools and your Alchemy skill. The important thing to note is: each player has herbs with differing poison effects, so herbs that make a poison for you may not make the same thing for another person.
Two herbs gives an effect strength of 1.0. If you add a third herb, you can mix a cocktail with up to three effects. You can also increase an effect up to 1.5 points.
NB: Not mentioned in the video, but one other way of losing the poison effect on a weapon is unequipping it. As a side effect of that, it is impossible to trade a poisoned weapon to someone else (and have it remain poisoned).