Help optimizing my laptop for gaming

There is nothing to really upgrade that will improve gaming performance.

Possibly the CPU, but only up to the fastest that Lenovo offers for that particular notebook. Just go to Lenovo's site to configure a Z580 to find out what is the fastest CPU which can be installed. However, most games are limited by the GPU not the CPU. Therefore, depending on the game there can be no performance increase whatsoever, to something that is a modest increase.

The best option of course would be to simply buy a new laptop, but whatever you buy, the performance increase must be high enough to justify spending more money. If I was in your situation, I would probably look at something with at least a nVidia GT 650m at the minimum. But I would likely opt for at least the nVidia GT 660m because I don't think going from the GT 635 to the GT 650 is really worth bothering.

Of course the cheapest way to improve overall performance is to lower graphics quality, play at lower resolution and playing in DX9 mode rather than DX11 which is more graphic intensive.... Assuming the game can be played in DX9 mode. For example, Battlefield 3 is not compatible with DX9 so it will not run in that mode.