To confirm what you already know... Remove any Icemaker you might have...remove the shelfs in the freezer compartment...then remove the back panel. This will expose the coils, which as you've already guessed...are iced up.
There could be two problems
1) low freon charge...causes more rapid icing of the coil
2) defrost cycle not working
2)a. faulty heater ( there is a heater below your coil. It's on a timer. It either looks like a glass tube with a coil inside...or like a metal element on an electric stove...except it's just a straigh bar that extends the width of the freezer.
2)b. faulty timer ( inside the fridge...you've got a timer that just goes in a continuous cycle. From time to time it goes into defrost mode. Switches off the compressor...turns on the defrost heater and the munion heater (usually near the bottom of the freezer compartment, inside the panel, next to the door opening)
2)c. faulty thermostat. There's a thermostat located near the coils to signal that they've defrosted...so the heater will turn off.
Here...check out this website...
It will help you diagnose..and if you supply your make and model, you can even order parts.
I work on my own appliances and have ordered from them in the past.