How can I connect my Guitar to the Computer?

If your amp has a 'line out', use that instead of the headphones port and connect that to your PC/sound card's 'line in'. This is, more or less, how you would connect an amplifier to a mixer. This time the PC and the audio software you're using is the mixer/recorder/every thing in between. Set the amp's output volume just high enough before any distortion or clipping occurs. Do all other adjustments from volume to effects on the software.