I had the same problem with my car this resulted in me going to a shop and having to get a new alltenator.
Check your battery connections. If they're tight and no corrosion is present, then your battery may be dead. Recharge or jump start the battery. Still clicking? Replace the starter.
You may have a loose or corroded ground connection. Not just the battery ground but the engine to chassis ground. You may also have a bad battery cable. If you see swelling on any part of the cable, replace it. Make sure the connections are clean and tight. Did you replace the starter solenoid with the starter, or is it a separate item? If the solenoid is separate from the starter, it is probably bad. It is unlikely that the ignition switch is bad, but possible.
The reason it's clicking is because your battery is dead. Replace the battery and you're good to go. It just isn't providing enough juice to turn the starter over correctly.
starter, solonoid, ignition, battery. if it clicks when you turn the key then the battery is really dead or its the starter the battery could still be jumped if its the battery.
check your battery, maybe it's time to replace it
Starter motor problems, charging system problems, wiring and connection problems between battery, alternator and starter. To many possiblities to list. If the battery is still charged and you need to jump the car to get it to start you have a bad starter. Now you say how can that be? it starts when its jumped? Correct the starter is drawing too much power"amps" and having a second battery "the jump" appx doubles the amount of amps to the starter allowing it to start. Replace the starter and problem will go away. Besides it wont take too long before a jump wont allow it to start.
Do you have any click in the starter itself but still not turning over? Then its the starter giving you fits. Replace it. If no click is heard then it could be a fuse or more likely the starter relay. Check them both before moving to the starter.
It could be your starter, or your starter solenoid.
Find an ammeter to find out what the starter is pulling. If the starter is pulling too much current, you probably need to replace the starter, but don't just go off replacing parts until you test them. Apparantly that plan didn't work to well for you when you replaced the battery and alternator. Always test first!
Double check the battery cables at the battery and at the engine, be sure they are clean and tight.
replace the starter switch on the steering column. advance auto parts part #cs96 about $11.00....
Check for damaged battery cables. It is very common for acid leaking from the battery to burn through the positive cable on a Cadillac. If you see any green where the copper should be, replace the cable. Also, do you have constant power at the starter at the 'big wire' terminal of the starter and power at the 'little wire' connection of the starter while turning the key?