Also, you don't want to feed an amplified signal into your recording device, unless your amplifier has a line output. You'd most likely use a stereo line output from the mixer to a line input your recording device instead.
I thought that too at first...but since he doesn't clearly indicate outputs/inputs....I think it's the other way around in the diagram...he's feeding the amp from the computer so he can monitor the tracks in the computer.
Are you 3 different people then Kang' or is that setup just for yourself and you intend to build tracks? If you ARE a group and intend to play together that is going to have to be one MOTHER of an amplifier/speaker rig! However, assuming that is so and you get a good sound from the speakers, all you need is a decent Audio Interface and a pair of mics.
But I suspect that is not your intention and in that case you can tear down that diagram and first of all read up on recording with an AI. Then you need to tell us what sort of computer power you have (but almost certainly good enough) and what the budget is.