There are no solid facts as to the origins of the Xenomorph species ; instead, there are many theories which cannot be confirmed. They may be an artificially created species, or they may have evolved naturally on a planet very different than our own.
In the film "Prometheus," it's implied that the Xenomorphs were created as biological weapons by the Engineers. To be specific, the android, David (an android built to be curious and human-like), infected Charlie Holloway with a drop of the mutagen and Charlie then impregnated Elizabeth Shaw creating a large Facehugger-like life form (aka hectopod or trilobite) . Elizabeth, in pain from the life form growing within her then removes the life form by entering the automated surgery table on board Meredith Vicker's exclusive ship. The life form later encounters an Engineer and seeds him much like a facehugger; thus, creating the Deacon which resembled the Xenomorphs.
It is heavily implied and revealed that in Alien: Covenant, David (Weyland's original Synthetic AI Cyborg) was the possible chief creator/developer of the first Xenomorph eggs that contain the "facehuggers". Utilizing the now emptied planetoid Elizabeth Shaw and David landed on, preceding Alien: Covenant, David was able to combine the limited number of creatures and organisms available with the "black-goo" (not to mention, unfortunately, Elizabeth Shaw's female reproductive system) to develop the first "face-hugger" eggs. David knew he needed Human hosts to eventually gestate the Xenomorphs to create his fantasy of creating something even better than what created himself. The Xenomorphs have become to be known as "the perfect organism" throughout the franchise. Thanks to Alien: Covenant, we now know that the Xenomorph is the end result of the creation cycle. --> Engineer --> Human --> AI Synthetic --> Xenomorph.
But, based on the Alien vs. Predator, the aliens are pre-existing and have been hunted by the ☀Predator. It is possible that the Aliens were annihilated in Alien vs. Predator: Requiem, but were resurrected by David in the Alien movie just before the were forgotten forever.
The Xenomorph could be the result of genetic manipulation by Mala'kak (also known as the "Engineers" or "Space Jockeys") as a terraforming mechanism about ten million years ago, but apparently it went horribly wrong. As a result, over the years, the Xenomorph has become an entirely new species in the galaxy. This theory has been twisted slightly by Prometheus, when the Mala'kak brought containers filled with an unknown substance (it closely resembles Xenomorph secretions in hives). This substance mutated harmless organisms into somewhat hostile creatures. When humans "infected" with this substance mate with one another it produced a different subspecies of "hecto-facehuggers" that can grow incredibly large. As shown in the movie, this new subspecies, regardless of who it infected, Mala'kak or human, would result in the "Deacon" or "Proto-alien".
Another theory, commonly accepted, touched on in the expanded universe, is they were the alpha predator of their own ecosystem on a nightmarish and harsh planet (some assume it to be Xenomorph Prime, but there is no solid evidence). Without their native ecosystem to keep them in check, they have since infested their own kind, and developed into a separate dominant species. Other parts of the universe began to take notice in this interesting species and began them. The Mala'kak, the Yautja, Humans, and an unspecified highly technological humanoid species introduced in the comic Alien: Reaper began using them as means of sport, terraforming mechanism, as biological weapons, food delicacies, and scientific experiments. 