iPhone Background Video
Cropping the video won't help, necessarily. What you need to do is render the video with the proper aspect ratio (roughly 9x16), and at or above the iPhone's native resolution, which is 750x1334 for the iPhone 6.

Source: iPhone 6 Screens Demystified

It's mainly the aspect ratio that is causing the problem you're seeing. Your video is probably 16x9, which will look fine in landscape orientation. For portrait, it needs to be rotated.

If you're filming footage, you could just rotate the camera on its side. If you're working with existing footage, you will have to crop the video and pull out a 9x16 "tall" rectangle out of your existing 16x9 "wide" frame. Any decent video editing program is able to do this. Unless your source footage is high-resolution (1080p or better), though, you'll have trouble getting sharp results, since you're essentially zooming into a very small portion of the frame.