here is the breakdown:
A. Moisture is very bad for pc's
b. A fridge (especially a mini fridge) is not made to run in a cooling state 24/7 much less for any extended period of time. A refrigerator cools the air inside it to a set temp and then shuts off. Once the temp rises to a pre-determined hot setpoint it turns back on and re-cools the air down to it's cool setpoint. The compressor inside a mini fridge will give out very quickly if you are evacuating air from inside the fridge. It would do even worse if you put it on a closed loop with your pc. Instead of ambient temp air you would be taking hotter than ambient air and pumping it back into the fridge.... Not good.
The only way to do this is to put radiators inside a fridge with 5/8 holes in the side of the fridge to allow for the tubing you would be using in a watercooling system. These holes need to be completely sealed. Put some cheap, easily replacable fans on the rad, inside the fridge, and watch the cool air work it's magic.
Even this method can fail! This is because, just like it was stated before, a refrigerator is not meant to cool something that generates heat. Name one thing that you typically put inside you fridge that generates heat. I don't know about you but i can't think of any sandwich i have ever crafted (no matter how awesome it was) that was so damn delicious that it radiated warmth. If you want sub-ambient temps go and buy a phase change system. Or take the components out of that mini fridge and build your own.
In response to your question... Even if you put a de-humidifier in the loop before the pc odds are:
1. It won't completely dehumidify the air
2. The fridge will break (quickly)
3. The dehumidifier will break down due to constant torture from that humidity generator you call a fridge
so the short answer (even though you probably read through the rest of this post to get to here) is that you cannot use a fridge to air cool a pc, it is possible (though fail prone) to use one to aid in watercooling though.
Sorry to be the bearer of bad news
now on the other hand... You could create a sealed system that routed air through a cooler parts of your house, like say a closed closet, or if you have an a/c vent somewhere in your room you could use that. The air coming from an a/c unit is typically much more dry than that of a fridge.