Hopefully you are replacing the alternator because you had a load test done on it, and it was proved to be defective. Otherwise, you may be replacing it unnecessarily.
When your battery is going dead, you need to first determine whether something is draining the power out of it when it shouldn't be.....OR...the alternator just isn't charging it back up. A load test of the alternator will show that the alternator is correctly charging, or NOT. If it is okay, then your diagnostics shift to finding out what is draining the battery.
In your case, a clicking relay can happen when the battery voltage gets to low and the relay solenoid doesn't have enough magnetic field to keep the relay locked. On the other hand, it might actually be the cause of your battery going dead.
They make inexpensive test lights that can help you diagnose what is draining the battery, IF that is the root cause of your problems (and not a bad alternator). Start pulling fuses until the test light goes out, and that will tell you the circuit that has something on it that is draining the battery. Electrical schematics also help you determine what all is on each circuit.
Respond back to this post with the progress you make.