Is capturing videos uploaded by some people illegal?

You'd be breaking some kind of copyright law, I'm sure. If you're recording the screen, though, there's no way anybody would know you've done it.

Except for looking at your browser history, but I don't think anyone is going to care.

Most likkely (unless it's the copyright holder that put them there) - but little to no chance of anything happening as long as it's kept "for your own use" - if you put on a disk and physically hand to someone you know, still little to no risk...

When you start really having any real chance of issues (and what they really look for) is if you are the one spreadding it all over the Internet, to people you don't know, and probably never will.