How do I make a video clip fit in this big green screen from a certain youtube video?

Use the 'corner pin' effect on the layer you want to be inside the green screen. Use the existing green screen as a guide to where to put each corner in the 'corner pin' effect. I would suggest putting each corner so that your new image is larger than the existing screen. Then key out the green screen layer using 'Ultra Key'.

Unfortunately the original clip has a rather poor track applied to the green and the shot is not stable ... so the result is never going to be great. You can try to manually animate the corner pin by keyframing each corner to track the movement of the screen and camera.

As Byron says - AE is much better at this stuff.

In After Effects you will surely have better results because you have many specialized tools to do this job.