Start-up based on PhD: inside or outside the university?
First things first, you need to find out whether you even have the option to do it independently of the university.  Since you have done your work at the university, the university likely owns at least part or possibly all of the rights to commercialization of your work.

Most universities have somethings like a "Technology Transfer Office" whose goal is to help turn university research into startups and/or licenses to existing companies.  These folks are often quite happy to work with you to help clarify the intellectual property situation and to help make connections that can support you in building a spin-out.