The mainboard is not powering up correctly -- the click you hear is a component attempting to power up.  The system will need to have the mainboard replaced, if you think it's worth the cost (in the $150-250 range).  You may be better off putting the funds toward a new(er) system.  
My Inspiron 3521 will not boot. When I start it, it only clicks constantly with no screen display and nothing I have tried will bypass this.  I have tried all of the usual boot options, F2, F8, F10, F12, escape, delete, to no avail. I replaced the hard drive but it just does the same clicking as with the original hard drive. I have power lights come on when I am trying to boot it but it will only click and the screen is black. I can hear the hard drive winding but no visual to see if the bios is showing.
Problem with Inspiron 3521 boot process