Is it a single click as you turn the ignition on or a series of clicks as you press the starter button? A series of clicks points to the starter solenoid being faulty. Follow the positive cable from the battery and you should find the relay (suzuki's are cylindrical in shape I believe) and see if the clicking is here. If so. with the ignition on short across the 2 bolted cable connections momentarily with a screwdriver and see if it turns over or starts. Careful, don't allow the short to touch the frame. Hope this helps.
my opinion , this should never get hot as you say it should always stay cool to the touch never hot if it is hot you may have a short somewhere ck the wires get a schemetic and find out where the wires are coming from and which one may be grounded out
if carbs are clean and not flooding etc then try finding out which cyls are off then swop plug caps over from good cyls and see if problem moves , if yes then fit a set of 4 new plug caps ! if no then do carbs again !!