Your primary way to keep in motion as a sniper is strafing. If you keep in motion - you're harder a harder target to hit, not only by opposing snipers, but by any shooting enemy. It's a lot easier to strafe out of the way of a bullet then it is to run forward/backward.

How Do I Strafe?
You should already have keys bound to +moveleft and +moveright. Make sure these keys are easy to get to while you have your fire button held down.

If you press these keys, you will move left/right without turning your body. This is very useful for the sniper so you can always look in front of you while sniping on the move. When engaged in a sniper war keep in motion otherwise, you're just a sitting target and most likely the first player an opposing sniper will aim for. In other words - A stand-still sniper is a dead sniper.

If you really want to be a harder-to-hit target, I'd recommend jumping also. Want to know more about strafe-jumping and jumping in general? Lets move on!