I used a company in Iceland booking a day trip. Was very happy small group. Small van . Great guide. I am looking for the same experience in Copenhagen. Specifically viking ships and going to Sweden .
daytrips only. Lorette
Some inspiration for day trips around Copenhagen.
I agree with ORDtraveler, it's very easy to do on your own. Trying to pay someone to drive you instead will cost a lot more, and most likely take a lot more time.
I have never used them and as a solo traveler they would be out of my price range, but if you have a small group, it may interest you.
They are called Tours by Locals. Ones I've read up on use public transportation, but if you want your own van, they can arrange for a vehicle and driver or will ride with you in your vehicle, but not drive you. They have set itineraries, but will also customize a tour for you.
As others have said, Copenhagen and surrounding areas look like they would be easy to DIY using public transportation.