Hi,

UK product?

I think the fridge is auto defrost, but the freezer isn't

Sounds like a cycle defrost....FF section frosts up and defrost itself
when the unit cycles off, the freezer is a manual defrost. These oftne
have an evaporator that frosts up between the walls in the back of the
FF section to cool the FF section, and -often- do not have a fan.

ice seems to build up at the back of the
fridge, but the whole fridge doesn't seem to get very cold.

Weak refrigeration system ( compressor, gas charge ), light staying on
inside when the door is closed, control not staying on long enough are
-some- possible trouble makers.

jeff.
Appliance Repair Aid
http://www.applianceaid.com/