As far as I know, there is a section of the copywrite law that allows you to use whatever you want for educational purposes and for practice. As long as you aren’t making any money directly from the creation of alternative soundtracks (using someone elses video) you should be all set. I’ve done this, and have had videos up on youtube for a couple years using trailers for different video games, replacing the soundtrack with my own.
that being said, I have an alternative suggestion for you.
Instead of using existing material, or videos, you can try downloading Unity or Unreal, and go through their tutorials to make your own super short game, and create the sounds necessary for it and integrate it yourself. That will do worlds more to show off your skills then just re-scoring something that already exists. If you can show that you can integrate the sounds as well, that will help others know you can do that part of the work. That and learning more about how the engines work is a very useful thing.