Usually, solenoid chattering implies that voltage applied to the solenoid's actuator winding is rising and falling , under load, at the chatter frequency. This can be because of several things:
1) the cable from battery to solenoid is not making good contact with battery or solenoid - remove, clean all contact surfaces with wire brush and then reconnect, TIGHTLY.
2) the battery does not have a full charge or is old/worn out. - fully charge the battery and try again. If it still chatters, jump to a known good battery and try again. If good now, battery is likely bad.
3) cable from battery to solenoid has corrosion inside the lugs or up inside the copper - put a jumper cable from battery plus to soleniod lug (in parallel with existing short cable). If starts now, cable is bad.
4) Ground cable's connection at battery of engine case is bad/corroded/loose or cable's lugs or copper has issues such as in 3) above. Remove, clean connections. If that doesn't work, use jumper cable from battery minus to engine case and try again. If it now works, ground cable is bad.
5) Something is preventing starter from turning (engine seized, starter motor seized or bearings need lube, etc.) and the starter is drawing way too much current - causing the battery voltage to drop and solenoid to chatter. Check for starter rotation with spark plug removed and/or try turning engine over (CCW) with socket on crankshaft nut - and lastly, remove starter and test on bench. Figure out which one of these is slowing down the starter.
If you get through all these and the problem still exists, then let us know - we'll have to investigate a little deeper... 99.9% chance that one of these will get it.