Eh. It depends on a number of factors. Are the zombies fast or slow moving? Or some of each? Are they simply humans infected with a disease (happened after the 1918 flu pandemic, look it up) to include bio warfare, strange parasites, nanites, or mutant rabies? Or all dead from any cause rising up everywhere? If diseased humans then what are the rules of engagement - ie diseased people now can spread TB or Hep yet you can't go around shooting them... Can the disease be cured, is it always fatal, how many people are immune, etc.
If you are trying to shoot man sized targets in the head, and trying to destroy the "essential" part of the brain, then you're talking about hitting a very small target under stress. (Targets to train snipers in this exist, the key point to hit is about the size of a dime.)
With a person suffering from sort of disease, assuming the rules of engagement allowed shooting them on sight, there's little reason to suspect that you'd actually have to shoot someone in the head. As in most things, destroying the brain would work more quickly, but they'd bleed out from a traumatic rifle wound to the chest, having a limb blown off etc.
A .308/7.62mm Nato rifle makes sense if one isn't going around playing at being light infantry. It will simply do a better job on man sized targets and cause more damage. (Modern 7.62mm ball ammo essentially duplicates the M2 .30-06 round fired in Garands during WW2 and Korea, an offering that no one seemed to complain about regarding its ability to deal with human sized targets.)
For less conventional origin zombies... Well it'd be expensive. You'd want a quality AR - LMT, Colt, Noveske, Barrett, etc. Then you'd want to modify it with several all important doo dads. The first is putting a registered suppressor on it and also owning another of the exact same model as a spare. No noise, no muzzle flash.
Next you get yourself a PVS-14 (yes other even more impressive sights can be had but they are even harder to get and fiendishly expensive, plus if zombies are dead a thermal sight might not work anyway). The almost 4 grand that you'll drop on your PVS, mounts, and accessories will essentially let you engage targets at night with a great deal of success. That your platform will be a suppressed rifle with the flash eliminated will make it all the more so.
You'll also need some sort of optic, ideally to work with the PVS 14. You could get one of the more expensive Eotechs (great for in close on moving targets), or try your luck with an Aimpoint, or get a top of the line Acog. Either way, you'll want to pair it with a dual laser aiming module. This will be on that has both an IR laser - that you see only with your night vision goggles on) that is zeroed to hit to the same point as a visible laser (for during the day and to simplify sighting the thing in).
Add ammo, mags, cleaning kits... A nice set of body armor with multi hit level IV rifle plates(because friendly fire isn't and humans will always be the real problem and not the zombies...) eh.... you're hitting about ten to twenty grand for your set up, esp once you figure in training.
About four grand will set a guy up with a "good enough" M4 clone or used Colt A2 type rifle, an Eagle universal chest rig, a set of Izzy body armor with rifle plates, and a used Kpot and put a nice Surefire and one of the better Eotechs on the carbine. That would essentially describe my problem solving set up, but I'm a frugal shopper and got it for less. Any old handgun would also be helpful. You can simply stuff it in one of the mag pouches on your chest rig.
