Instructions
Vampire
Find a vampire to fight. You can find them in many caves, tombs and ruins. This includes Aralen Ancestral Tomb, Nerano Ancestral Tomb and Serano Ancestral Tomb.
Cast the spell "Weakness to Common Disease" on yourself. This will increase the chances of becoming a vampire. This is optional but helpful if you currently have high disease resistance.
Let the vampire hit you with punches. You will eventually receive a message saying that you caught "Porphyric Hemophilia."
Wait three days. Your character's race will change into a vampire.
Werewolf
Follow the main quest line in Solstheim until you are in Skaal. This island can only be accessed if you are playing the "The Elder Scrolls III: Morrowind: Game of the Year Edition."
Fight the werewolves in Skaal. You will contract a disease called "Sanies Lupinus." The attack on Skaal and catching the disease are both mandatory parts of the quest line. Do not cure the disease when the NPC tells you to do so.
Wait three days (in game play). Your character's race will change into a werewolf.