It could also be the neutral safety switch. Mine has been going out for a year, every once in a while, I have to step on brake and shift it in and out of gear a few times and it will start. just my .02

 

Did you charge the new battery? Try to jump it? When a starter does not have enough power it just keep clicking. If it has enough power it just clicks once and hums pretty loud for several minutes before you either drain the battery or fry the starter. .

If it is just click, click, click it is either a defective starter or low current at the starter. The reason the click over and over again is because you turn the key the first thing that happens is the battery powers up the solenoid which is a large switch, that allows power to get to the starts but because there is not enough current to charge the winding in the starter the voltage drops so low that the solenoid does not have enough power to stay engaged and release the contact plate (the switch) once it releases the switch the voltage is no long trying to fill the starter winding so now there is enough voltage to power the solenoid again and the cycle continues. There is nothing else anywhere in your car that can cause the clicking condition except and short in the starter windings or low voltage to the starter.
Step one check the voltage at the battery. On a new fully charged battery you should have around 12.6 volts.