Welding to copper is not easy because copper has very high thermal conductivity. Aside from that there is no real issue. You could braze it, depending on what temperature range you need to withstand. This is also called "Silver Soldering" - melting point is in excess of 600°C so pretty high. Anyone can do this with a propane or MAPP torch and the appropriate flux and solder.
Soft soldering (what we do in electronics) doesn't work well with most thermocouples, but it does for coppper-constantan, so that's a possibility if you don't need to go too high in temperature (the thermocouple itself is usable to 400°C, solder melting point is typically lower). I'm not suggesting this, but you could even connect just a constantan wire to the copper and use a type T signal conditioner (not suggesting it because the copper you have may not be sufficiently pure to not require calibration, not because it won't work as a thermocouple- sometimes they add tellurium or other impurities to aid production procceses)
To avoid problems, you should use a measurement device that works with a "grounded junction" thermocouple. Any battery-powered device will be fine as will most industrial type thermocouple transmitters and signal conditioners (they should say they have "galvanic isolation"). It's also possible to use differential (instrumentation) amplifiers, but I would suggest avoiding it outside a very controlled environment.
Remember that thermocouple signals are relatively low level (tens of uV per K), and if you have a lot of EMI floating around you could have issues. Also the temperature of the junctions inside the measuring instrument must be measured to greater precision than the desired system accuracy because (to a first order) thermocouples measure temperature differences. That means that high absolute accuracy may be hard to achieve, but seeing tenths of degree C change in a lab environment is easy.
The best accuracy will be achieve by using a fine wire gauge since heat necessarily flows in or out of the wires. For most purposes you can say as fine wires as practical is best.