What was it? How did he become such a fine marksman? Was he a military man? If it was mentioned in the movie I must have missed it obviously. The movie is set about 10 years after the American Civil War so was he a veteran of that conflict? Quigley said he was from Wyoming. Not many people in Wyoming now, and during the ACW even much less. I guess he could've migrated there after the war but that is never mentioned. Not a major point, but I wonder about things like that.
"check the imdb cast list before asking who portrayed who in movies please"
I'll take a stab at this, although bear in mind what I'll say is more anecdotal than based in any documented facts.
In the American West and Midwest, from the time people first started migrating that way until the present in some ways, shooting guns is a way of life. I grew up in rural SW Nebraska and most kids learned to shoot as a matter of course.
Back then, shooting was part of survival - if you lived out in the West and Midwest, you'd need to put food on the table with your gun. Any kind of big game would be taken with a large rifle, such as a 45-70, 45-90, or even what Quigley used - a 45-110. Being a good marksman would open up the number of shots you could take because you can't always get close to game animals like deer, elk and moose.
Aside from that, marksmanship was a valuable asset that was often tested in various shooting contests and exhibitions, so that may have also played a part.
For what it's worth, I wouldn't see military training for the Civil War to be a place that would focus on fine long range marksmanship because they were still using antiquated block battle formations, so it would have been less about accurate individual shooting and more about mass volleys.
I can agree with your angle about Quigley's shooting background. Makes perfectly good sense since it is not revealed in the movie at all. But in difference to your statement about the use of antiquated block battle formations in the ACW, those formations and tactics were the state of the art at the time. And sniping, although in it's infancy was used quite a bit. But most of these guys brought their shooting skills and their rifles with them to the army at first. They learned their "trade" in the way you described trickg1, at home shooting for some food to put on the table. Good answer.
"check the imdb cast list before asking who portrayed who in movies please"
The Sharps was used a lot by Buffalo Hunters who supplied meat to the men building the railroads. Buffalo Bill Cody was one such man. Later the buffalos were slaughtered for their pelts and to deny a food source to the plains Indians so they would starve to death.
Buffalo were so placid that if you hid in the tall grass you could kill many with a single shot rifle like the Sharps at a considerable distance. it was like shooting cows.
The buffalo hunters killed themselves out of their jobs and had to find other work.
The Civil War made considerable use of snipers, some of whom used some pretty accurate weapons for their day, particularly compared to the standard junk the military issued.
Quigley was certainly the right age to have served in the Civil War, and might well have been selected for a sniper based on his obviously VERY superior eyesight and perhaps pre-existing shooting skills.
It's also possible that he was born in the American frontier and, like many, felt no particular obligation to get into what they viewed as an Eastern war. Certainly many, if not most, of the frontiersmen had great shooting skills just to put food on the table and protect themselves and their family. Considering his rifle and his long-distance skills, it's also entirely possible that he made his living, at one time, as either a scout or hunter for either the Army or the railroad. His choice of weapons (extremely expensive compared to most) would certainly be ideal for hunting elk or bison ("buffalo") -- or bear, for that matter.
We obviously can't tell from the evidence we're given, but logic seems to suggest that some combination of the above backgrounds would be a good bet.