So, canonically, it is uncertain whether or not the Xenomorphs actually eat, or if they are just powered by their blood, but there are certain examples in the movies which we can refer to to determine whether or not they actually eat.

• In Alien we don't see much of the Xenomorph's behavior beyond it attacking. It can be noted that it does kill with its headbite, which might be considered eating, but we can't be sure.
• In Aliens, we see the long dead remains of the colonists, which don't seem to be damaged by Xenomorph consumption, however, most of them are embedded in the walls of the hive, so it may be hidden.
• In Alien³, we get the first canonical evidence of a Xenomorph eating. There is a scene during the foundry chase where one of the inmates discovers the Runner Xenomorph feeding on the body of another inmate. Also, Golic says that the creature "feeds on minds" although this is most likely how he viewed a headbite through his delusions.
• In Sea of Sorrows it is noted that the hosts of the Xenomorphs had been consumed.

Non-canon depictions

• In AVP: Requiem Xenomorphs were shown feeding on the bodies of their victims
• In the Aliens: Labyrinth comics, we are shown a look inside of a hive, and, if memory serves me right, we see Xenomorphs feeding on people.
• While not the adults, there have been several examples of Chestbursters consuming their hosts after they emerge in the comics.
• In the novelization of Alien and, I believe a scene that was intended to be in the film, there's a scene where the crew finds out that the creature was in the pantry and ate food.
• In the Aliens vs Predator 2 videogame, there is a part in the alien campaign in which you eat a cat, although you are in the chestburster stage at this point.

Conclusion

Even with all the evidence pointing towards the fact that they do eat, it is important to note that Xenomorphs can last without food for an incredibly long time, which is evidence that, if they do need to eat, they don't actually need to while in hibernation.