Why is my fridge building ice on the outside but not getting cold inside?
That is a sure sign that the refrigerant has leaked out. There is just enough left to freeze over the first bit of the coil as the remnants flash vaporize after passing through the expansion valve due to high suction. If a meter set was connected, one would find low high side pressure and high low side pressure. 

If it wasn't a high priced refrigerator or is out of warranty, don't bother with a service call, just replace it. Do not attempt to repair it youself unless you have a EPA 608 or 609 license depending on if it is a R410A or R134A unit.