It depends on the guitar amp and the sound card in your computer.
What you do not want to do is to plug a speaker output into the sound card...it will toast it pretty quick!
If your amp has a Line Out, that will work. Then you would connect it to the Line Input on your sound card.
I'll warn you now though, that unless the output on the amp is compensated (that is, has a high frequency roll off to mimic a speaker), it's going to sound like a bunch of mosquitoes.
The safest (and aurally, the best sounding) thing is to put a decent mic in front of the speaker on the amp.
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How to play the blues on guitar