The range limit for killing a deer or antelope with a .308 will be determined more by your skill than anything else. With the right bullet put in the right place it will "kill" (not necessarily dead on the spot) a critter out past half a mile and probably much further, if we're talking just the effect of bullet.
In terms of placing it there, without having to do lots of trajectory and wind figuring, using a reasonably accurate rifle and a 200 or 225 yard zero, I would say about 300 yards. Beyond that the trajectory drop becomes very abrupt (the .308 is not a particularly flat shooter) and wind becomes very much harder to figure.
Work on your fundamentals like consistent gun mount to shoulder, cheek weld to the stock, aiming, trigger control, and position shooting (prone, sitting, etc.), how to use a shooting loop sling (Ching or military M1907), get some experience with range judging and crosswinds, and you will be able to hit a 10" paper plate at 200 and 300 pretty darn consistently. That should be a dead deer. If you can keep all your shots at least 1" away from the outer edge of the plate, that would probably do for antelope as well.
Either the 150 grain or 165 grain bullet will do the job. At 300 the 150 will shoot a little bit (an inch or so?) flatter than the 165, and the 165 will have about that same amount less wind drift than the 150. Boattails will help but only by a fraction of an inch, so, pay your money and take your choice.
Like Jack said above....PRACTICE! That is the most important thing.
