Someone please explain to me how this Amp Business works.

This. It will work but you should look into a source upgrade next.

I'd also consider getting a different amp.

nuforce will do everything you need.

But can it play Crysis?

Could be an starting place but it's never.... ever... going to drive the K70X well.

The best way to get audio off your computer and into an amp is a digital to analogue converter (DAC). Your computer already has one actually, in its sound card, but the stock ones often aren't that good which is why many people favour an external DAC or soundcard.

ETA 2 : Many amps you can buy have an inbuilt DAC as well, it might be worth looking into some.

ETA - and why wouldn't a LD IV drive a K70* well? I don't understand that statement at all.

An amplifier does just that, it amplifies the sound output from your source. In this case, your laptop is the source.

There are a number of ways you can hook your amplifier up to your laptop; you could buy a 3.5mm to RCA cable to connect the amp to the headphone out on your laptop. In this case your laptop sound card would be converting your music - which is in digital format - to analog sound output which your amp amplifies.

It's likely your laptop sound card doesn't provide a 'line-out' sound level and is providing an already amplified sound output (as the headphone-out is typically used to drive headphones, obviously). For this reason you need to be careful of the output volume from your sound card when driving an amp (for one it could result in the output from the amp being too loud to give you a use-able volume range on the amp, and second I've heard too loud an output can damage amps in some cases).

Sound cards are not generally regarded as an optimum source to drive a high quality amp and headphone. Sound card DACs (digital to analogue converter) and amps are often sub-par, resulting in loss of detail and an audiable noise floor (as they're internal to the computer often resulting in EMF interference, etc.), etc.

A dedicated, external DAC is a popular choice because it essentially replaces the sound card and does a much better job of converting the digital music on your computer to an analogue sound output - thanks to better conversion hardware and a dedicated, external power supply, etc. Plus an external DAC typically provides an analogue line out to connect your amp to. DACs can be hooked up to your computer in a number of ways depending on the interfaces available on your computer and the DAC. Typical interfaces are USB, optical, co-axial, etc.

You can also buy combined DAC and amps. There's a wealth of information on this site about good amps, DACs and which makes and models of these components pair best with the headphones you have.