Make a video and post it on Youtube and don't worry about it. One of two things will happen. The artist/record company will take it down - this rarely happens anymore. Most often they'll put an ad on your video and an ad on your video page where the listener/watcher can buy the original and they get the money. You cannot monetize a cover video because it's not your copyright.

A few years ago this was very different but today most labels are actively encouraging covers on Youtube because of the exposure and extra money they make from them. And technically you are supposed to have a very expensive sync license to do covers on Youtube but honestly no one seems to care anymore and it's rarely enforced. As a matter of fact from my research if you are a popular cover artist who gets the views labels will come courting you and it's even possible to make a few dollars for yourself with a deal to split some of the money. But it's very hard to achieve that status.

Josie