Personally, I'm biased against 15" laptops, and Numpads, and HP in general. With all that said, considering your budget, that HP laptop is probably one of the better options in that price range (for gaming).
But don't get that one you linked on Amazon. The graphics card in it uses shared memory and isn't that great, and the low-res screen isn't worth it. Those can be upgraded on that model. Check out this configuration:
Customize your HP ENVY 15z-j100 ($675) You can use coupon code SVD8492 to take another 5% off. The 8750M is a lot better than the 8650 that's in the model on Amazon. And it has the 1080p screen.
I don't know anything about the build quality, I'd just look up video reviews if you're interested. At that price point you're going to be sacrificing something for sure.
I think there's a pretty big jump in quality going from $600-$700 to $800-$900. Look at the Lenovo y410p and y510p. If you want something more portable, check out the Acer Aspire V7-482PG. With all of those you'll get a nicer display, a much more powerful laptop, and a better quality screen.