Hey Willard, your right in that if I build a typical case it would not do the job... I'm planning on building the case around the hardware so that it draws the heat away from the components. Copper is great at conducting heat away. The more copper, the better the heat is conducted away. The trick will be to design the case that draws the heat out and doesn't insulate it.
Perhaps I will design the case as one big radiator with fans and fluid...
I really don't think it's possible to cool through the case. It might look awesome, but I don't think you can expect any real functionality.
The big problems are as follows:
Heat conduction is about more than just using a good material. The rate of conduction is also proportional to the mass of what you're conducting the heat into and its cross sectional surface area. Additionally, the temperatures are going to build up incredibly quickly in the vicinity of parts you're cooling because the amount of heat conducted is inversely proportional to the distance from the hot spot. Unless you use low end parts and underclock them, you're still going to need fans to actually dissipate the heat because it's not going to get very far from the components in the first place. They'll just keep getting hotter and hotter until they fail.
CPU coolers have solved this problem by putting the radiator right on top of the CPU (obviously not what you want, since you want the whole case to be the radiator), or by using heat pipes to move the heat quickly. A heat pipe is not just a copper tube, it's also depressurized, filled with liquid (usually water) and coated with a wicking agent (commonly copper powder). As a result, the water boils quickly, moving heat down the pipe where it condenses on cooler surfaces, dumping its heat and is finally wicked back to the hot spot to repeat the cycle. This is clearly not something you can build yourself (loss of depressurization will cause the heatpipe to drop to nearly 0% efficiency immediately).
What's more, you've got to balance the ability to absorb lots of heat with the ability to dissipate it. The rate of dissipation varies inversely with the ratio of volume to surface area (meaning big chunks of copper don't dissipate heat very well). So if you get enough copper in contact with the CPU to effectively absorb the heat, you'll have simultaneously created the problem of not being able to get rid of that heat.
It's just not feasible to spread the heat out through the case in any way even approaching evenly. It's also not feasible to simply use a large hunk of copper to absorb the heat without some other method of dissipation, like a large radiator and/or fan directly on top of the hot spots (also known as a traditional CPU cooler). On top of all that, the thin walls of a case are going to be really, really bad at conducting heat because of their extremely low mass and cross sectional surface area compared to an actual CPU cooler.
This may seem counterintuitive, but it all has to do with the difference between temperature and heat. You could build this case and see the temperature quickly spreading the case and think "hey, this works great!" but you'd be missing what was really going on. A large mass requires much more heat than a small one to increase its temperature by the same amount. So even though the large mass will be at a much lower temperature, it has absorbed much more heat, and results in much lower temperatures where it matters, in the component.
The laws of thermodynamics just aren't on your side here. There are serious engineering problems here, and I don't mean to sound condescending, but I doubt you've got the physics or engineering background to tackle them. You can't just slap a chunk of copper onto a hot component and expect it to work well.