How to get started on a PhD?
In The Netherlands, as a PhD Student (promovendus) you will generally be a salaried employee of the University with a 4 year contract (which may be terminated at the 1.5 year mark if things are looking bleak). On the website of the University Association of The Netherlands you can download the CAO which contains information on the regulations under which you'll be working (articles 2.3, 3.1, 6.8 and E.12 are of particular interest). If you are working part-time then the length of your contract would reflect that (e.g. you'd get an 8 year contract if working 50%).

You can also do a PhD as employee of a company (buitenpromovendus) in which case these regulations would not apply. These generally take far longer (as you are performing regular work for the company, and your work on your PhD is tangential to this).

If your tutor has offered you a position, then the application process could be as simple as saying "yes". There is no formal admission process, though you must have a Master's degree in the relevant field before pursuing a PhD (if this is not the case, then it is still possible but more complicated).

In The Netherlands, applying for a PhD is the same as applying for any other job; if your tutor has offered you a position, then all you need to do is accept. If the offer is not yet formal (or contingent) then a normal application process would consist of submitting your CV and then (hopefully) getting a "job interview".