Tipping a private chef is recommended based on a percentage of the total cost of their services. For most private chef services, a minimum tip percentage of 15 to 20% is suggested. I understand that you're asking this question because you appreciate the service you've received and want to do the right thing. The right thing here is whatever you want it to be. As Dario clearly states, there is no tip requirement.
You couldn't leave a tip and be okay. However, if you personally want to show additional gratitude, simply choose an amount that fits your budget and your idea of what you want to do. In most cases, private hotel chefs will already come with the tip indicated on the final bill. These may consist of an 18% service charge.
If your chef does everything possible to provide you with an unforgettable experience, you can add a 5% tip. Depending on the performance of the private chef, you should always consider giving him a tip. The industry standard tip is 15 to 20% of the total bill. However, it can also vary depending on your level of service.
For a contracted private chef, 15% of the total cost is the minimum amount of tip.