So typically on a refrigeration unit, the most common problems for not cooling are:
1) Compressor running, but not cooling - this usually means that your sealed system sprang a leak and if you keep running it like that, the compressor will burn up. The fix is to locate and seal the leak, evacuate the system and recharge. Not typically a DIY thing and less expensive just to replace the freezer. Place your hand close to the compressor, if you feel it vibrating, it's running.
2) Compressor not kicking on - Either the compressor is fried, the thermostat is not functioning correctly, or the start relay isn't working correctly. If this was the case, you wouldn't hear any noise at all coming from the unit. Since you hear something, I would guess that this is not the issue unless there's a fan for cooling the compressor and that's the quieter noise that you're hearing. I don't think that having a fan on a freezer is very common since the condenser is all along the outside of the skin usually.
Does that help? My guess is no freon. I've never checked on a chest freezer, but on my newer mini-fridge and my vintage GE refrigerator I can hear the freon flowing through the evaporator as it cools. If the compressor runs, and you hear nothing flowing and get no cooling, that's probably the issue.