Is downloading music videos an infringement on copyright laws, and if so how would I go about getting permission to use them

Yes, playing videos for commercial use in public is copyright infringement if you do it without the permission of the owners of the copyright of those videos. You can try to find yourself who the copyright owners are and contact them for a license. You can also contact companies that specialize in this business. For example, the Copyright Clearance Center at www.copyright.com does this type of clearance and licensing work. This is not an endorsement of this company but rather is meant to point you to alternatives to clearing this yourself, which could be very difficult.