QueueHammer, I'm assuming you're from the USA.
You're going to need a really, really good car for this.
Driving from, random example, Dijon to Stockholm (that will knock-off three or four countries for you) is an incredibly long trip on the world's best and fastest autoroutes (and also a few thousand kilometers on Sweden's tidy laneways).
(1) To do this you will need a car which surprisingly nobody else has mentioned.
(2) You're going to need almost certainly a diesel vehicle because you'll save a little on fuel costs, and (on the whole) they're realistically better for autoroutes.
(3) A huge tip is that in France in particular, they have a very advantageous long-term car rental scheme for tourists from overseas.
most importantly, you'll get the sexy red "tourist plates"...
and again the price is a steal. It's pointless outlining the scheme here as you can google thousands of pages on it.
I recommend a Renault, all modern cars are the identical quality and Renault's pretty much the biggest and best represented. So it's likely a diesel Renault.
(*) Footnote, generally, forget about the idea of buying a car. Back in the 1970s, for example, to tour around the USA you could easily "buy an old car", use it up and then get rid of it. It's simply impossible with the paperwork, etc, in Europe of today. Forget it.
(*) Footnote, if you're rich, as you probably know you can buy a new BM or Benz built-to-order and collect it in Germany, have a long holiday, and they ship it stateside for you. If you're rich, phone them today and do that. Setting that aside, you're best bet for a long-term auto holiday in Europe is the French long-term tourist leasing deal.
(4) Where to arrive? CDG is the most hateful experience this side of hell, so try to fly to say Lyon and get your vehicle from a Renault dealer there.
(5) You're itinerary!
Remember! don't go widdershins!
(5A) Here is your itinerary A, all the good bits...
So, Lyon, Monaco, Bologna, Venice, Graz, Prague, Munich, Switzerland (say, Arosa), Geneva and back to the Renault dealer in Lyon.
This will give you all the best in the world, and a few adventures, and very few boring bits.
Note that it is really extremely not-that-interesting to drive from say the South of Germamy (which is utterly awesome if you love food, mountains and attractive women) to say Copenhagen (which is utterly awesome if you love food, and attractive women) ... so, "South of Germany to Copenhagen", for example, is incredibly boring and takes a long, long time. So itinerary A above carefully avoids all the boring bits and still gives you a "I drove all around Europe!" feel.
If you want more? Here's Itinerary B ...
(5B) Here's the completely baddass "I drove a long way" system:
Fundamentally, Europe is a political entity around a very large lake (the Baltic), so do that. So roughly:
Lyon -> Bologna -> Adriatic countries -> Eastern countries to Tallinn -> Scandinavia (it's such a small word to write) -> the low countries -> Paris -> Bourdeaux -> the two different parts of Spain with Portugal in the middle -> and back to your friendly Lyon Renault dealer. (Note: although not indicated on the map, be sure to work in Bavaria (southern Germany) as much as possible when in that section.)
Lyon, Bologna, Tallinn, Lappland, Paris, Porto, Lyon.
It's the LBT-LPP route!
So - that's how you do it.
(*) Forget about the UK, it's not part of Europe in most touristic certain senses, and the food's not European.
(*) You could do this in a campervan if you're very uncool, but it seems to cost as much anyways to park those at night.
(*) It cost me 120 euros to fill the family car last time. So in terms of "How much will it cost?" the answer is, quite simply, "an incredibly high amount of money".
(*) Be sure to take in a couple of the alpine passes ... for example
(*) Amazingly there are still some car-trains in Europe. Hop a car train for the leg Hamburg-Basel for instance.
(*) You should begin and end at the same place. It's cheating if you don't. If you do not begin and end at the same place: for the rest of your life people will scoff at your achievement and say, "oh, you didn't go around in a circle, whatever dude..."
(*) Note that Europe has many excellent restaurants. Take plenty of money for that too. Allow 100 to 2000 a day.
(*) I forgot - because Europe is enormous, you are dealing with issues like crossing the arctic circle, as well as the aforementioned mountain passes on one of the planet's larger mountain ranges. (And I guess extreme heat in the south, if that's an issue for you.) So, the time of year you undertake this big drive, is a central issue. Cheers!