Time/hours put into gaming is a good way to gain (G). As long as you don't spend too much time on playing multiplayer on games that either have no achievements or achievements you've already got. That eats up a lot of peoples' time. Playing the same shtuff
over and over again. I'm not going to encourage buying and playing the easy (G) games, as they are all mostly crap, but there are some decent ones in the bunch.
Assassin's Creed II (good, but you might not be able to 1000/1000
Need for Speed: Most Wanted
Call of Juarez: The Cartel (you can get roughly 700+ Gamerscore in about three days)
Here are a few games off the top of my head, even though I don't understand why you
need Gamerscore quickly. Presumably there's a reason though so hopefully those games will help you. Some of them might be a bit hard to find though so eBay, Amazon, and stores which stock pre-owned games are the way to go.
Sports games. I would go NCAA and Madden. Most but not all are easy 1000. I think between them both you could get about 10,000 pretty easy and not cost you that much. Those games drop in price every year after the new one comes out.