To see whether it is cost effective to use such a system, after you invest in all of the hardware you need, you should look at the Latent Heat of Vaporization for CO2. I've only checked one resource and it listed the value as 241BTUs, about 70 watts, for 1 lb of dry ice. I'm pretty sure that the ratings for most CPUs are watts/hr, and if you're looking to put your whole motherboard in a fridge, you'd have to add up the heat from the chipset, video card, hard disks, etc.. So it looks like you'd be looking at at least 3 lbs of dry ice per hour. Not sure if it's cost effective, but it would be good exercise.
I think the issue that must be addressed is avoiding any condensation. Therefore, your refridgerator/freezer has to be sealed tight, and probably evacuated before adding the dry ice. (You couldn't exactly have a separate condensor inside the fridge for moisture, since the whole fridge would work on the same principle.) Otherwise if any moist air gets into the refridgerator it will freeze and remain frozen until you run out of dry ice and it warms up, probably quickly since you'd have your complete system inside.
I'm not sure what your use of this PC will be. If it's for getting the ultimate overclock for a few minutes, it could probable be done, but if you want to play BF2 for several hours on dry ice, it would probably be easier and cheaper to buy a faster CPU and GPU or do water cooling.
If you're deadset on using dry ice in some way, and money isn't as much an issue, I'd do a closed loop cooling system with chilled gylcol, using dry ice to cool the heat exchanger/radiator. You could plumb this all inside your fridge and condensation in the fridge wouldn't matter. However, you'd need to insulate and then heat wrap the cooling lines going to the CPU heat exchanger in order to prevent condensation.
I've had a few years experience fighting condensation using heat wrap tape when I worked as an engineer, and have learned not to have much faith in it, so be forewarned.
If you do choose to make a dry ice system, good luck and please share your experience.