We know the Xenomorph from Alien must eat something since it is a living thing and one of the primary tenets of living creatures is they need energy to live and grow. It is illogical to assume the alien Xenomorph does not eat ANYTHING and yet can grow to well over six feet tall and weighing at least two hundred pounds.

In addition, the creature has an armored, acid-proof carapace, as well as pints of hydrochloric or hydrosulfuric acid for blood, copious amounts of mucus-like drippings, able to spin catacombs from fluids in its body, able to sprint at speeds of twenty miles per hour, and has the strength to rip through body armor. I find it hard to believe it can do this on a single meal of human heart, blood, liver, kidneys and assorted sweetbreads.

Since we have never seen them eating, we have to infer from what is in their surroundings, what they could consume to continue to put on mass, muscle, armor and blood to become the formidable killing machines we know them to be. We can use the first two movies for our extrapolation.

Alien

In Alien, there is a single creature and all it has to eat are the humans on board the Nostromo and anything they brought with them. It was shown that Ripley found an empty storage container during the hunt for the creature. So, it can be assumed, it needed mass to grow and in this case was able to eat the food of its host creature to survive. This seems reasonable if the creature has taken some of its biology from the host, ensuring at least some level of compatibility with the organisms ecosystem and food supply.

It can also be inferred there was more than one storage container of that type on the ship so it does not take much to assume human food, plus the humans it killed during the hunt were sufficient for it to get the mass it needed to grow and expand in size.

All of its advanced biochemical processes as well as its silicon makeup would probably be able to be met by the technical equipment that made up the ship in the form of metals and other inorganic materials found in a starship of the size and complexity of the Nostromo. Unless a thorough scan of the ship was done, it would have been able to eat its fill of inorganic material and it would never be noticed or missed by the crew.

Aliens

In Aliens, we have almost the same conditions, a sizable human base, plenty of human food stores, a significant number of humans to eat (and drink from, blood is a decent source of iron and lesser minerals) and refined inorganic materials in great abundance. So their growth, size and numbers can be presumed based on the amount of food available. We can assume they may be biologically efficient, processing a greater amount of energy and converting more mass than we do, so it may not take much organic material for a Xenomorph to develop physically, which is why they are so willing to abandon their host if the nymph finds itself in danger.

In summary, Xenomorphs eat what their hosts eat and can likely find a source of inorganic material compatible with their biology on any planet they find themselves on (Silicon is the eight most abundant material in the universe and the second most abundant mineral compound on Earth). This high level of adaptability would surely make them an Alpha predator on any world they had sufficient compatible inorganic material for them to complete their molting and grow to their adult size.