Game guide: Making a Zombie
The first Method - Bitten by a Zombie
Be bit by a zombie. In some cases, the side effects will only last temporarily, like in cases when a Sim is turned into a zombie using a Flask of Potent Zombification acquired through the alchemy skill. If your Sim was bitten by a zombie that rose from the ground after nightfall on a Full Moon, your Sims transformation is permanent unless cured through elixirs gained from the alchemy skill.
The second Method - Potion Making
Through potions made at the alchemy table, there are 2 zombie potions you can make. A zombification elixir can be made at Level 3 Alchemy and can be used to make a Sim temporarily a zombie and any Sims they bite will have temporary zombification too (Both can be cured by an elixir.), the other elixir is Potent Zombification which can be made at level 10 and found through reading books, it makes your sim permanently a zombie (unless cured by an elixir). Anyone bit by that zombie becomes a permanent zombie too (Unless cured by an elixir earlier mentioned.)