Bloodthirst first started off a strain of blood that the Lilin Fang dipped into a poison that was supposed to kill Morbius. Instead Morbius survived and had the lilin bloon in him.
For a while Bloodthirst would toy with Morbius's mind to kill anything. When Morbius was first killed by Switchblade, Bloodthirst took over Morbius's body until he was killed by Switchblade as well. When Morbius was brought back to life, Bloodthirst was still in his mind. As hard as Morbius tried to resist, Bloodthirst would end up taking over Morbius's mind and siding with Lilith and the other Lilin.
While Bloodthirst now made his name known while in Morbius, he killed Louise Hastings and he lead the Lilin to Doctor Strange's mansion where the Midnight Sons were hiding. Doctor Strange later helped Morbius take control over his body again, keeping Bloodthirst dormant.
Later on Morbius would get rid of the Lilin blod and keep it in a separate becker. Soon the becker would explode and Bloodthirst would take his true form and get revenge on Morbius.
He attacked Morbius at St Judes Hospital where Bloodthirst wounded Morbius's friend Jacob and his boss Mandy. Morbius would fight Bloodthirst only to be close to being defeated. Morbius then took a bunch of needles containing something that can destroy blood cells and stabbed them into Bloodthirst's neck, killing him.