How to record guitar covers through my amp?

In order to get the best sound quality, you'd want an USB interface that would connect the line out on the amp to the computer. You might have difficulties if you have built in overdrive or distortion, however. If you want the video, you could use a webcam (with or without an internal mic) and record a video, while recording an audio track into a program like Audacity, Krystal, or if you have a Mac, GarageBand. Then, you could get a program to separate the video from an original audio track if necessary, then sync the good guitar track with the video. It's complicated, but it can be done.