Car rental in Copenhagen is very easy. You will find all the major car rental agencies in both the city center and in the airport. You can rent cars from: Sixt, Hertz, Europcar, Budget and Avis. So if you prefer a specific agency (or if you have a deal), you can find them in Copenhagen.
In Copenhagen Airport all the agencies have car rental in the corridor between Terminal 2 and 3 (the international part of the airport). When you are arriving and have got your luggage, just turn left and walk about 50 meters. Then you will see the car rental agency signs.
Avis, Europcar and Hertz are also represented in Terminal 1, which is the domestic part of the airport. If you want to rent a car from Budget or Sixt, you can just catch the airport bus to Terminal 3. It is free and it only takes a few minutes.
In the city center you can find all the agencies. They are all close to the Town Hall Square, the Main Station, the Tivoli Garden and a lot of the hotels.
It is not cheap to rent a car in Copenhagen; especially if you are used to rent car in the United States. But it is actually possible to rent a cheap car in Copenhagen.
Often the car rental agencies have special weekend offers. Long term rental is often also cheaper. You might find a car for a week for the same price as 4 or 5 days. Another way to get a cheaper car is to prepay it; instead of renting it when you arrive.
You will probably also find the car in Copenhagen smaller than in most other countries (in particular the United States). But the distances to everything in Denmark are short, so you will not need a large car; unless you bring a lot of luggage and your whole family.
If you prefer a big car, it is possible to find all the European luxury models in Copenhagen. That includes BMW, Audi and Mercedes-Benz. If you like to spoil yourself, rent a cabriolet in the summer. Or make your parking easier with a BMW Mini Cooper.