Winchester does sell white box ammo, I wasn't much impressed with the accuracy from it though. Plus it was pretty high up on the $$$ meter.
Basically, the main thing you need to check is the bore. Shine a light through from the muzzle end after removing the bolt. You want to see nice sharp rifling, and no blackish areas, just nice shiny metal. There isn't as much to look at externally for a shooter-grade rifle, the standard I use is, "Would I accept wood in this condition if it was a baseball bat?" LOL!
Also check the muzzle to see if has been counterbored. If you can see that the muzzle has been bored out, it's a possible sign that the barrel is getting worn, but the acid test is, again, taking a look down the bore.