A few things need questioning before we get to that.
Are xenomorphs genderless? More to the heart of the question, who makes the queen?
I imagine the aliens are asexually reproductive, and one becomes the queen every cycle or so to breed the eggs. The eggs grow into facehuggers, then chestbursters, and then full-grown xenomorphs.
So then, I assume a few of new full-grown xenomorphs turn into a queen?
The comments to your questions contain good resources about the rest of the life cycle.
To answer your specific questions:
So then, I assume a few of new full-grown xenomorphs turn into a queen?
As we see in Alien 3, a xenomorph is either a queen or a drone already in its chestburster stage.
In the "Assembly Cut" of Alien 3, we even see a special "queen facehugger" that is presumed to be predestined to impregnate a host with a queen chestburster. We don't know if this is considered canon, although it has also appeared in novels and comics.
In the cinematic version of Alien 3, however, the facehugger that laid the queen chestburster was just a regular facehugger. In that scenario, it's possible (although pure speculation) that it could detect that there were no other xenomorphs in the vicinity, and so "decided" to make a new queen to create a new hive.
Are xenomorphs genderless?
We don't know. We never see a xenomorph "inseminate" or otherwise interact with a queen before she lies eggs. However, again in Alien 3, we see a regular xenomorph (although different from the ones from the previous films, this is implied to be because it gestated inside a different host species) accompanying and protecting the host of the queen xenomorph. Perhaps it was born to inseminate the queen as well, but that is again speculation.
