How to approach supervisor about running a start-up along side PhD?
First, make sure this is allowed at your institution.  Then develop a solid plan for how you will handle both projects before taking it to your supervisor.  To the first point, it seems that you already have, but make sure that you understand your institution's rules.  For instance, is it allowed to use school resources?  What does that mean in your institution?  It could mean something as obvious as using a school computer to run a simulation for the business, or something as subtle as taking a phone call in your school office.  You need to be 100% sure that you are not violating any institutional policies.  

Some institutions are hands-off when it comes to external projects as long as they aren't done on campus, others aren't.  For instance, where I did my masters, we were not permitted to make income outside of the school without losing an equivalent amount of funding.   Where I did my PhD this was not the case.

Next, make a clear plan for your advisor to understand.  As was pointed out in the comments, both projects are going to take a LOT of time to do well.  Your PhD should be the priority.  You need to make sure that your advisor understands precisely what working on the business and working on the PhD mean.  Be realistic.  It's very likely that you will check your business email while working at school, so don't pretend that you won't.

Your advisor has already stated that you should keep your side projects to your free time.  You need to be very clear what that means.  PhDs aren't necessarily like jobs.  You might not be able to get much research done one morning because the ideas aren't there, then work well into the night to make huge progress.  Does that unproductive morning count as free time? 

Have a strong plan for what you want from your advisor. "I want you in on this" is not helpful.  "I need a support person in the role of algorithm development.  Here are the specific responsibilities I was thinking.  What do you think?"  is much better.  

Finally, make it clear how much freedom the advisor has in this business.  Is he a co-founder, making equal decisions?  Is he an employee?  A contractor? A board member? Will you pay him?  What happens if he decides to hire a few co-op students and bring his long-time collaborator on board?  Who retains control of the business?  Who has ownership?  These aren't unique questions to student-advisor businesses, but they are important to get straight up front.  The worst thing that can happen is you both have different ideas for your responsibility and the business gets away from you.