This was a horrible, horrible mistake.
Attempt one, I used waaay too much. The computer booted, then crashed horribly as the heat was transferred from the chip directly to the motherboard by the oozing mass of goo.
Attempt two didn't get through the boot because I scraped off too much and it didn't actually transfer heat.
Attempt three looks like it's working so far... *knocks on wood*
Man, there's goop everywhere smeared over bits of my computer and, like, my body. Why is it a paste? Why not, like, something solid that would melt or whatever?
Anyway, it looks like the constant crashing was *probably* not becuase of the thermal paste at all -- my computer was set to be overclocked somehow. When it said 'press F1 to go to setup, press F2 to use default options' I hit escape, which apparently did neither. This time I had it use the default options and in particular NOT overclock, and it's been stable so far.
Of course, I had three stable days even when it *was* crashing, so that doesn't necessarily mean anything. I might still be doomed.