This is a nice older simple unit, I have a similar one. I'm going to assume you have a multimeter and basic understanding of electrical parts. If not feel free to ask for more details.

Do you have interior lights, fans, anything? Does the compressor try to start and shut off or nothing at all?

1) If the freezer and condensor fans are running you may want to just skip this step and move to rear of machine. But you can try turning the defrost timer manually---It is behind the cover at the top of your unit where the knobs are. Pull knobs off and the light shield. Over toward the right will be a hole with a white/tan one directional stub you can turn with a flathead or fingertip. It only turns one direction. It'll click as it turns and may go a whole revolution or more before anything happens. Turn it until you hear a distinct louder click, that is either start or end of defrost. If the freezer fan stops that's defrost. If the freezer fan starts that should be compressor run, fans will start, etc... If none of that happens you have a bad timer or burnt out timer. Here's a video on how to expose defrost timer, if not exact it'll be similar. You do not need to remove the metal cover for what you are doing now.
GE WR9X489 Defrost Timer - AppliancePartsPros.com

2)Check for voltage at the compressor in back bottom machine compartment. If it is not in defrost (freezer interior fan and condenser fan do not run in defrost) It should have 120v AC at the compressor. You may actually smell a burnt component when you get down there. I'd start with the unit unplugged and look for obviously melted black plastic. If no smell and no visual damage you'll need to see if the compressor is actually getting 120v.

If you have running fans, lights etc...but no compressor action I'd guess it's either the start relay, overload, or capacitor (less likely). Do the overload and relay together if possible.

Relay - GE WR07X10098 Start Relay - AppliancePartsPros.com
Overload - GE WR8X172 Overload - AppliancePartsPros.com
Capacitor - GE WR55X24065 Kit Capacitor 15 Uf - AppliancePartsPros.com