You're pretty much on your own as far as repair goes. There is very little info available on the net on repairing and recharging a fridge.
Since the system is sealed, you will need to add service ports. You can get piercing ports that clamp onto the lines and then pierce them. Note these should be removed after service since they introduce a possible source of leaks into a sealed system. You can also braze standard service ports in. You need to pull a vacuum on the system and then charge with nitrogen up to maybe 50psi or so and use a soap solution looking for leaks.
Brazing should be done with silver solder. Do not use soft solder, it will fail.
If you find the leak, there's a couple ways to fix it. You can find a heat sensitive epoxy stick that you melt over the leak, that works for small leaks. Brazing works too, but only on copper lines. Worst case, you may have to replace a component like a condensor or evaporator. Note that many components are aluminum and may have a thin copper plate on the end for joining. You need a light touch when brazing those. When brazing, you need the system filled with an inert gas like nitrogen or argon to prevent formation of scale inside the line which will screw up the compressor.
Most residential refrigeration units use a capillary tube as the metering device. Any crud in the system will block that which is a major pain.
Some (or a lot) of the lines may be buried in foam. You may need to dig that out. Foam can be replaced with the spray insulating foam available from building stores. Use the window/door stuff so it doesn't expand and destroy the refrigerator.
If you are in the US and the refrigerator uses R134A, you can use those little cans available for auto A/C recharge. If you do, make sure that it's only pure R134A and not one of those with stop-leak. You will need a gauge set. The refrigerator should list the type and weight of charge, so you'll need some way to weigh what goes in. Refrigerators do not use a lot of refrigerant and they are pretty picky about having the correct charge. Overcharging can burn out your compressor. Undercharge and it won't work very efficiently.
If it uses another refrigerant, you're probably SOL. Those are only available in larger cylinders and their sale tends to be regulated to licensed contractors.
Do not vent refrigerant, you can be seriously fined for that (tell that to all the backyard yahoos dumping refrigerant into leaking car A/C units)
Refrigeration is not rocket science, but more like a medieval guild profession with most information closely guarded and not easily available.
If you pursue this, by the time you buy all the tools and supplies you need, you will probably spend 2x the cost of a new refrigerator