You will probably have to spend over $1500 if you want a laptop that plays modern games at highest settings with no lag.
If you turn the quality down, you can get away with a lot more. My brother's computer has integrated HD 4000 graphics, and it plays LoL at medium-low settings at 60fps. On highest, it plays it at about 20-30fps.
You only 'need' 4GB of RAM for gaming. You can get away with it, 8GB is preferred, but if you have a budget that is always something to consider. More RAM =/= proportionally better performance in almost all cases, unless you are playing games while doing other RAM-intensive tasks. The highest RAM usage I have ever seen in any game I've played is 3.7GB, so if you were doing other stuff at the same time, that would cause issues with only 4GB. If not, there would be no problem. It depends what you are doing.
As for the processor, quad core at most will net you better performance. There are no games that use 8 cores, the highest I've seen is 6 at one time. I should know, I have an 8-core processor and I keep track of it during gaming. The first 4 cores are the only ones I ever get use out of.
ASUS is a good brand with very high quality products. However, I would instead recommend to compare performance not solely on reviews, but on GPU benchmarks like
this one.>>3 The savings are minimal. You can get a 1TB vs. 750GB for less than 10 dollars more in some cases.