The Sniper is one of the most significant long ranged characters in the Fallout universe. I say that because one shot one kill is just so much more effective than emptying an entire box of 5mm ammo out of your mini gun.
Character Concept and Creation
Character concept is a very stressed idea in any RPG. A very important aspect of your character is how he will be Role Played. This should be decided well before you enter further creation. The Character concept behind a sniper will likely be limited by your GM. While sometimes a military or ex-military unit is possible chances are you will be trained as a peace officer for a city or you are a proficient shooter out of necessity. Being part of a backwater town and hunting is more likely the source of your skill than being an ex member of the Enclave who left because of morally disagreeing with what they were trying to accomplish.
The race of your character should be logical. While I said a tweaker shouldn't be a min-maxed freak, he should be able to do his job. The human is the very base of a tweaker character. The human can be used in so many ways to get the job done. Snipers can be pretty much any race that can shoot a gun. Your best sniper will be a human although you could bring in some interesting back story if your character is 200 year old Ghoul sniper.
traits aren't really important with a sniper. Of course bloody mess is a great one. Otherwise I try and avoid picking a trait with my sniper. I get uncomfortable knowing that I'm going to get less perks or lessen my combat ability if I go too trait happy.
The statistics of a sniper are usually very similar with differences being in the areas of luck and Charisma. Make sure your character can handle himself and shoot well. You wont need high hitpoints or amazing strength but you do want to make sure you can wield weapons with high strength requirements as well as take multiple shots per turn to improve your chances.
The Skills tagged for a sniper should include small guns, sneak, and outdoorsman. I include outdoorsman because a higher outdoorsman skill sets you up for a better position in a random encounter. It can save your ass from getting mauled by a horde of death claws.
A sniper will want to carry a couple different guns effective at very long range as well as a good amount of ammunition. Something I like to have on my sniper characters is minor first aid items. The wastes are a depressing place and many adventurers are junkies, including snipers. Jet, Mentats, or Buffout almost always make it into my snipers pack.
The development of a sniper should depend on the type of campaign you are in. If you don't do much across the world traveling you might want to min-max a little bit and avoid pumping too much into outdoorsman. Always make sure to get your small guns and sneak to a decent level. You may be at a great range but enemy snipers are good at watching for you. Ask your GM to adjust rules to outdoorsman in cityscapes or on local travelings if you feel it could add dimension to your character.