There could be a number of reasons for this problem you will need to work through them from one end of the system to the other to diagnose.
Start with a bit you know is working.

1. Look at the receiver unit first. Do the indicator lights come on when you switch the microphone on? If yes hold the microphone belt pack about a couple of feet from the receiver and move it from one side to the other. Do the lights flicker? If it is a 2 channel (dual diversity) model -do the A and B channel lights flash on and off to indicate that the receiver is switching channels? If they do the problem is unlikely to be the microphone or belt pack.
2. Is your amplifier/ computer/ audio system producing sound from other sources (e.g. radio etc)?
3. Unplug the output lead from the back of the receiver and plug a cabled microphone or other sound source / instrument to it to test the cable.
4. It may be some sort of external interference. Most of these sort of microphones can be switched to different frequencies. Try changing frequency (usually a little rotary switch on the back of the receiver and inside the battery pack - will often need a small flat bladed screwdriver to operate it) and make sure that there is a clear line of sight between receiver and transmitter units with no large lumps of metal or electrical devices in between.
5. Do not use re-chargeable batteries as they almost never achieve an adequate voltage. Always use Duracell Ultra or equivalent.
6. The capsules and cables of lavalier / lapel microphones are very prone to failure. Try removing the microphone jack lead and plugging in a different one . If you don't have another microphone try a simple audio lead - maybe a mini jack-plug to standard 6mm jack and connect an electric guitar or other sound source (or just touch the bare plug contacts at the other end of the lead to see if it creates a sound) . Does it work? If it does then it is possibly the microphone itself rather than the belt pack. These microphone capsules are not repairable and mending the cables / jack plugs can prove very fiddly, (lost are moulded units and cannot be dismantled anyway) however replacement microphone capsules are not very expensive (about £16-£20 in the UK) but before you replace it see steps 7, 10 and 11 below because it could just be a problem with the jack socket on the belt-pack
7. Look at the jack socket itself. These often attach to the case of the belt-pack with a threaded ring that screws onto the socket unit. Sometimes the screw ring can work ever so slightly loose - which means that the jack plug doesn't quite go far enough into the socket to make a proper electrical contact (we had exactly this problem with a different brand of lapel microphone).
8. If the tests at no 6 don't find the problem examine the wire aerial on the belt pack carefully. Sometimes the central core of the aerial can get cracked by frequent bending (usually when people stuff the belt pack in their pocket). Try twisting the wire backwards and forwards gently. Work all the way along it's length - does this make the unit work -albeit intermittently?
9. Dismantle the belt pack to check that the aerial wire has not got broken off the circuit board.
10. While you are in there check that the wires to the microphone jack socket have not got broken off or stretched so that the screen and core are shorting out.
11. Put a battery in the belt-pack and try touching the contacts on the microphone input jack socket with a small screwdriver or other metal object - does this produce any sounds? If so it may be that the contacts inside the socket are bent and therefore not touching the pin of the jack-plug. Some careful re-bending with a thin probe may solve the problem.
12. If you have got this far you can be fairly certain the fault like in the microphone capsule because you have eliminated every other possibility