As we mentioned before, trying to actually measure the voltage across the wires will be very difficult for most people, which is why we strongly suggest using a thermocouple interface chip. The nicest one we've seen so far is the MAX6675 (and its replacement version called the MAX31855) which unfortunately is only available in SOIC package. While not too difficult to solder, we nevertheless have in the shop a breakout board that is ready to go .

First thing to determine is which wire is which. As you recall, thermocouples are made by welding together two wires, the chip reads the voltage difference between the two. One is the negative (for K-type its made of Alumel) and the other positive (ditto, Chromel). Luckily the wires are color coded, and almost all of the time you'll find the Alumel is red and the Chromel is yellow.

Connect the leads as required to your amplifier: