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 the odd few thousand km 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 millions of pages on it.
I recommend a Renault, all modern cars are the same 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 - in the way that in the 70s, say, to tour around the US you could "buy an old car", use it up and dump it. It's just not possible with the paperwork etc, in Europe, 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. 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) and takes a long time. This itinerary carefully avoids all boring bits and still gives you a "I drove through all the heart of Europe!" feel.
It would be nice to add Spain to itinerary A, but it's a bit out of the way.
(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 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.)
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 certain senses, and the food's not really as good.
(*) 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 pretty much "a lot".
(*) If you can, 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. And when you subsequently tell people about it for the rest of your life, they'll scoff and say "oh you didn't go around in a circle, whatever..."
(*) Europe has many excellent restaurants. Take plenty of money for that too.
(*) 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 an issue to consider. Cheers!