[Harmless Teddy Bear]

The question of why a lich would need a vampire is key.

As a food source or as a food provider?

Maybe the lich could be something like the Deadman in the Garrett Files. Dead and unable to move yet mentally awake and with telepathic/telekinetic abilities.

[NicknamedBob]

"The question of why a lich would need a vampire is key."

That depends on what story you want to tell, and who you want the bad guys to be.

It is very mainstream today to have the rich industrialists be the heavies. Of course, you could have that just be a lifestyle and cover story for the real bad guys.

Any group of immortals striving to remain unseen among a populace would tend to develop bitter rivalries. Depending on your sympathies, one group can be regarded as the good guys.

... regarded as the good guise.

Sorry! I got distracted. I do also like the food chain analogy. You could even intimate that there are other forms either parallel to, or even above the ones described. Sentient or not.