At some point in their life (gestation?), force sensitives gain midi-chlorians in sufficient quantities
Yup, everybody's favorite part of Star Wars: the midi-clorians. And we learn about them in everybody's favorite movie: Episode I: The Phantom Menace.
All living cells have mid-chlorians in them.
"Midi-chlorians are a microscopic life form that resides within all living cells." –Qui-Gon Jinn
And it is through midi-chlorians that life can exist and connect with the Force.
"Without the midi-chlorians, life could not exist, and we would have no knowledge of the Force. They continually speak to us, telling us the will of the Force. When you learn to quiet your mind, you'll hear them speaking to you." –Qui-Gon Jinn
The fact that midi-chlorians are the basis of life is affirmed in the StarWars.com Databank entry for the "Force Planet" that they originated from:
Not marked on any star map and not recognized with any specific name, the Force planet is a mystical world and the foundation of life itself. It is the birthplace of the midi-chlorians, microscopic organisms that connect the living Force to the cosmic Force.
Having lots of midi-chlorians means that you are stronger with the Force, and is characteristic of the Jedi.
Obi-Wan: The reading's off the chart...over twenty thousand. […] Even Master Yoda doesn't have a midi-chlorian count that high!
Qui-Gon: No Jedi has.
Putting this together, everybody has midi-chlorians in their cells (that's why there is life), but you are only force-sensitive if you have them in sufficient quantities. The more you have, the stronger you are in the force.
Given that being strong in the Force is hereditary (e.g. Anakin is strong, and so are his children Luke and Leia, and grandson Ben), I'm going to use Occam's Razor and suggest that it is their genetic makeup that determines the number of midi-chlorians their cells can support. Just like how a child's hair color is determined from the genes of their parents, so too is the amount of midi-chlorians that their cells can support. This is similar to the genetics of magical ability in the Harry Potter universe.
Because midi-chlorians are necessary for life, it seems that they are present when life begins. The simplest explanation of how they get into the cells is that they enter from the mother during the gestation period.
To answer a few of your questions:
Do the 'Ones' choose who has the Force?
There is no evidence that the Ones grant Force abilities to every force-sensitive person in the Galaxy, even if they have the ability to do so with people who they actually meet (genetic engineering?). Given that most people are unaware of their existence, but midi-chlorians are well known, it seems unlikely that they are hands-on with every force-sensitive person in the Galaxy.
Luke inherits his Force ability from Anakin, but what is the case when a Force sensitive child is born into a non-Force sensitive family?
Likely that they got the genetic makeup that resulted in a high number of midi-chlorians (see similar answers to How do Muggle-borns end up with magical ability?). Note that despite being twins (same parents, same prenatal environment), Luke was much stronger in the Force than Leia. This suggests that Luke's genetic makeup resulted in cells that could support more midi-chlorians than Leia's genetic makeup.
Also, how does that Jedi holocron in The Clone Wars tell the Jedi where the Force sensitive children are?
That's unfortunately a mystery. However, the fact that one can scientifically examine the number of midi-chlorians (Qui-Gon did it through a blood test on Anakin) means that it is theoretically possible for some advanced technology to detect midi-chlorians at a distance.