The 'clean' sound you get from a Twin isn't really 'clean' it's not at all similar to the sound being output by your pickups - the sound from the guitar is shaped by the impedence of the input, the colouration of the preamp and the tone controls, and the acoustic response of the speakers and the cabinet. (this is setting aside any additional colouration you get from overdriving the inputs - compression, distortion and breakup effects). This is one reason why direct line-ins sometimes don't sound great, even if you have a quality preamp or interface. The other main reason is the effect of mismatched impedence between the guitar and the input (as mentioned)
So if you want the canonical "clean" sound, you have to mimic these effects. The easiest way would be to run the guitar signal through either software amp and speaker simulation (like Amplitude, as mentioned, or dozens of others), or use an external modeling box like a Line6 Pod, Korg Pandora, M-Audio BlackBox (also doubles as a USB audio interface) or one of dozens of models of Digitech, Zoom, and others.
Or go old school with analog boxes like a Tech21 SansAmp plus some compression and EQ.
The advantage of getting an external box is you can also use it live or away from the computer, and they have a gazillion different modeled sounds, dirty and clean and echoed, etc.