You need to know the protocol (serial connection settings, message format) of your adapter. Not sure if the manufacturer documented message format for this adapter. I suppose you could use a sniffer (e.g. wireshark) to find out the format of messages yourself. Your code would be very similar to the co2 sensor code, connections settings and the message exchanges would be different.
2) Connect your thermocouple to LiV Pi via I2C (there is an I2C port available on LiV’s back). You can use a TMP513 chipset, have a look at the schematics here:
http://www.clever4hire.com/special-articles/home/a-thermocouple-measurement-circuit If you use this chipset (or a similar one), you need to see the message format over I2C bus so you can write your code. Your code would be somewhat similar to LiV LCD code. You should also make sure that there is no I2C address conflict with BMP180, LCD and DS3231 (for TMP513, address is B8 so that should be OK).
If I find an open source project or code sample, I will add the link here.
Once you get the measurements on LiV, you can write them into LiV DB, add a new graph, trigger alarms, etc...