No, fleas do not die after biting humans. On the contrary, after feeding on blood, an adult female flea will go on to lay eggs, usually directly on the skin of its host whether that be a human or an animal.
After a flea in its pupa stage emerges from its cocoon, it has about two weeks to obtain a blood meal from a warm-blooded vertebrate or else it will die.
Once it has successfully obtained blood, it is considered to have reached the adult stage of the flea life cycle and can then go for many months without feeding again if necessary.