Liquid cooling a PC on liquid metal?
While on the surface this might seem to be a good idea, in actuality, this is a very bad idea.
There are two metals (not including alloys) that are liquid at room temperature: Mercury and gallium.
First off, mercury is extremely toxic and should only be handled by experts.
Gallium will corrode aluminum and steel, which is what the coolant runs over/through to sink heat. It will eventually destroy the joints and heat sinks, which will lead to the next problem.
Both mercury and gallium are electrical conductors. If either of the two liquids were to leak onto the electronics, it could cause shorts and even damage the electronics. And again, mercury is extremely toxic. This alone is a reason not to use them.
Mercury and gallium have a high rate of volumetric expansion due to heat. Under high heat, they can expand greatly and the pressure would destroy the cooling lines.
Gallium itself isnt a liquid at room temperature. It has a melting point of 85.58°F (29.76°C), which means of the PC was turned off and it completely cooled, gallium would solidify. This of course could cause problems, since the liquid would not be able to flow.
Editing in some more thoughts:
Mercury is very, very heavy. One liter of mercury weighs a hair under 30 pounds (13.5 kilograms). One liter of gallium weighs 13.02 pounds (6 kilograms). It would take a massive pump to move that liquid around. The weight alone could cause PCBs to flex or break.
Everything in Keltari's answer is right, I just want to expand it with some other important info:
When you want to "transfer" heat, you need to deal with 2 major values: Thermal conductivity and heat capacity. First is how easily get/give heat from/to other material, like get the heat from hot surface and give the heat to cold surface. The second is how much energy can it store.
Thermal conductivity of liquid metals are very low compared to solid ones. Pure, solid, aluminium has a thermal conductivity of about 200 W/(m K), pure copper is about 390 W/(m K). Mercury, on the other hand, has a value about 8.5 W/(m K) and the value for water is about 0.6 W/(m K). So liquid metals are better than water for heat transfer, but much worse than solid metals.
The heat capacity is another part. A 1 K change in temperature (i.e. 1 °C or 2 °F change) for liquid water requires 4.187 kJ/kg, while the same change for mercury is 0.125 kJ/kg, this means same heat from the CPU surface incurs a 32 times larger temperature change in mercury!
If we think simply, 14 times better conductivity and 32 times worse heat capacity is about 50% worse sum related to water cooling, and still not taking into account other dangerous factors, like toxicity or the short circuit factors. (This calculation is not proper, because there's many other parameters which these values depend on, such as current temperature, pressure, and there is side dissipation on transfer, etc.)
Liquid metal CPU coolers already exist:
http://www.guru3d.com/articles-pages/danamics-lmx-superleggera-review,1.html
This one uses NaK : a eutectic alloy of sodium and potassium, that is frighteningly reactive with air, water, and just about anything:
https://en.wikipedia.org/wiki/Sodium-potassium_alloy
The same alloy is used for cooling in the nuclear power industry.