You'll want to use the line out to get the signal to your computer. Use a shielded cable, not a speaker cable. You'll probably want an unbalanced cable (TS not TRS). It's the one with only one black stripe.
YOU MUST HAVE A SPEAKER OR DUMMY LOAD CONNECTED to the speaker out, though. Sorry for the CAPS, but it's important. You will most likely blow your output transformer if you run it without a load. You need a dummy load that matches the amp's impedance and can handle the wattage if you're going to run silent. The
tutorial on the site covers it in a little more detail.
The line out on the K-Tre is after the power amp, which is nice because it means the signal will include any power amp saturation, which is a critical part of the amp's sound. So, in your case if you need to buy a dummy load for silent running, you don't need to worry about it having a line out, and you don't need to be as concerned about it rolling off the lows which some people (including me) complain about with the cheaper boxes because you'll be getting the signal from the line out on your amp. I think your biggest concern will be that it can handle the wattage 'cause that's a mighty powerful beast you got there.
If you're not concerned about totally silent running, you can put your cab somewhere where the sound from it won't interfere with your EQ decisions or piss anybody off... like in a coffin, buried in the back yard. :-) Or, you could use an attenuator, as opposed to a dummy load, which will shave off some volume, but won't be silent.
You can run the line out from the amp into the Great River if you're not getting enough gain, or if you want to include some transformer saturation in your sound.
Hope this helps.