It's possible to finish the game without killing anyone, but you most likely won't avoid combat completely. There are random encounters on the world map which force you to start combat. Of course you can run and don't kill anyone. You won't benefit much from Gambling skill. Unless you want to completely destroy the game economy. :) Barter can be useful only in the beginning. Later you'll have plenty of caps and items for sale. Science also doesn't play a huge role in the game. You can skip on this one as well. ??? Invest in Small Guns, Energy Weapons and Speech skills. These are basically the most important and you should finish the game without a problem. Also it's good to take Gifted trait. The advantages are worth sacrificing some skill points. As for SPECIAL stats, try to boost Agility as high as possible. Your Action Points depend on it.

Sarafan told you most everything you need to know. But yeah, avoiding combat is going to be a bit more of a challenge. I'd recommend you get familiar with the game first and then try a run like that. Also Agility is something you should not skimp on with your SPECIAL stats. Intelligence is really nice too. Charisma is mostly worthless.

Check out PIH (beware of spoilers, though.)