What is the proper way to ask a professor to be your PhD supervisor?
I don't think that your approach is not good. 
It is always preferred to be straight to the point with direct focus on your aims and how would they fit into your potential hosting lab vision and goals.

Always, Keep It Short and Simple (this is called KISS principle).

However,
Some suggestions to enhance your odds:

0- Set your objectives and your target areas.

1- Get prepared by identifying your potential advisor' research interests by visiting his web site and his lab activities to see if it's fit to your needs.

2- Make a balance sheet between your strengths and weaknesses.
Don't be shy to identify your weaknesses, because you shall be providing an improvement plan on how to address them.

3- Try to come up and suggest some research topic that sticks with your potential advisors research interests.

Like this, your advisor will be more than glad to offer his/her expertise and embrace your proposal.

I believe by doing the aforementioned steps, you show that you are already on your way to a PhD. Such thing would be visible to your potential advisor and would be certainly appreciated.

Good luck.