The short answer is practice. I suggest taking a valiant, and sniping a foot-soldier heavy faction on hydra's LZ, or london's balcony; it's good for learning these shooting styles:
-following the shot with just the scope
-following the shot with both scope and character strafing (both analog sticks)
-have the enemy walk into your waiting crosshairs by predicting where, and when an enemy's head will pass them (dragoons are prime targets for this)
You'll also have to learn to gauge enemy movement velocity to do any of this.
Get good enough, and you can even do some wicked shots like headshotting an enemy behind a wall that broke line-of-sight with you for just under a second without needing a thermal scope to know where their head is.
As for the close-up moments where you don't have as much time to do those things, you'll have to get used to quick-scoping as someone mentioned. Center your screen on your target, scope in, pull up a bit, shoot. If you know you can send the target into a stagger/combat roll with a shot to the chest, go ahead and bait that action.
SI is perhaps the training-wheels sniping kit, but damn effective at making the valiant a one-shot-kill against many things with energy drain.
I play on PS3 using the stock controller.
Again, it's all practice. I've been mostly sniping since demo, and I'm at the point where it's more effort to put how I snipe into words than it is to just do it. When you commit those styles to reflex, and not thought, along with earning 20 headshot medals before 25 kills by wave 3 without needing to stagger-cheese with energy drain/sabotage, that's when you know you're ready.