Sounds like you're using the integral temp controller. On a fridge, that should work and save the $$ on an external.
Without knowing your model and photos it is difficult to say exactly how to relocate it. If the sensing bulb is integral to the thermostat then your worries are few.
If that's the case, there should be merely 2 wires going to the t-stat unit. You would just need to extend these (using at least the same gauge wire) to wherever you want the thermostat to be. Watch out when screwing new mounting hardware because the refrigerant lines are easily pierced and hitting one will ruin your weekend. If you have no idea what I'm talking about here then photos would be helpful to post.