Whenever I think about counter sniper, two different scenarios come to mind:
1. You are set up in a position and a crafty enemy sniper locates you; an engagement begins.
2. Although there are countless ways this can happen such as a enemy sniper hide location being radioed in to you or stumbling across a snipers location on your own, you are spotted by an enemy sniper and chose to engage the target.
In general I find that sniper vs. sniper engagements either last forever or are over before you know it started. For the first example, you can either try and out shoot your opponent or you pick up and move when you think the situation has gotten too dangerous. If you fail to knock out your target initially and you are forced to 'haul ass out of there' the tactic I most commonly use is simple; disappear only to 'show' yourself again in the most unlikely of places and take a shot. Try and leave eyes on your target(I would use a spotter), and hook around to that unlikely position while in radio contact with your 'eyes on man', pop up and squeeze off your shot. Hopefully you will only have to do a relocation tactic once in your efforts to take out the enemy sniper but sometimes that is just not how it goes.
While out scouting, stalking, or crawling to a position and you locate an enemy sniper there are a few things I like to do. If I am uncertain on a possible target I like to send out a scout and lie in wait to see movement or a shot taken at my advancing scout. If I have time, I like to set up into a well concealed position and simply wait and watch for movement or some confirming object of the enemy sniper. If out of the corner of your eye you notice movement and you 'just know' it is a sniper the key to success is to say calm and professional; remember all your skills: get eyes on target engage quickly or if still unnoticed find cover and engage.
If you have been sent out to knock out an enemy sniper element, gather as much intel. on the surrounding environment and location of that enemy. As usual proceed with caution throughout your stalk but be especially careful when you are getting close to the possible location of your target. When you have a visual of the general area the sniper was known to be operating in, I find that I rarely move more than 20ft every 5-15min. You must take a very long time at each position you move to scanning the area. I am glad to have a spotter with me because a two man team can always have one person scanning while the other moves. Once you have a confirmed visual on the enemy sniper, the goal is to get only as close as you need to take a confident shot at the target; you don't want to try and get too close and be detected nor do you want to compromise your position by taking a shot from a distance you are unsure your rifle will be able to easily hit from.
The general idea of counter sniping takes all of the basic, and not so basic aspects of sniping and intensifies them to the limits. After all, the biggest threat to any given sniper is an enemy sniper.