As outlined in this article/log by Marius Taciuc, the thermocouple in the HAKKO clone tips seem to be type C. I did some measurements of my own, by boiling a cup of water, sticking the iron in the cup, together with a cooking thermometer and plotting the thermoelectric voltage against the measured temperature. 

Even though my methods weren't as good as Marius', I came to the conclusion that I could expect about 19uV/C.

Since all I had at home was N-channel mosfets, I was planning on doing lowside switching and using a differential amplifier for amplifying the voltage of the thermocouple. How ever, I couldn't get this to work. My theory is that the the voltage was simply to small for the LM358 to accurately amplifiy. I tried my best to match the resistors for the differential amplifier, as well as using opamp followers for the outputs, but nothing seemed to help.

Instead I tried using a simple non-inverting amplifier. This would require a high side switch (more on this in a separate log) since it required common ground, but this way, the amplificatoin seemed to work without issue. The initial plan was to make sure that the amplified voltage was 1V/100C. As it turns out that requires some very precis resistor matching.

 I settled for for 500k  (1M // 1M) and 1k, for Rf and R1 respectively. for an approximate gain of 501, depending on the exact values of the resistors. This give me about 0.95V/100C. I can compensate for the discrepancy in software.