Once again, the extravagant, effusive praise of vatrader feeds my voraciously hungry ego, compelling me to reach beyond my grasp regarding expertise in diagnosis and repair, and to make recommendations beyond my actual competence, denying my real and profound limitations!
Just a brief tutorial (follow along on the excellent wiring diagram accessible fom vatrader's link, class!).
The starter circuit relay is an "enabler," it enables the control voltage from the starter button to reach the solenoid connection of the starter relay, closing the high-current starter relay contacts and powering the starter motor.
The starter circuit relay itself is activated by the logic of the safety switches (e.g., neutral switch, sidestand switch, clutch switch).
So, we've gotta energize two relays to crank a stock KLR650; starter circuit relay, and starter relay.
The, "CLICK!" must be, I'd imagine, one or the other relay solenoids engaging its respective contacts. If the contact surfaces provide sufficiently low resistance, the control voltage is passed on and ultimately closes the starter relay contacts, connecting the battery positive terminal directly (no fuse here, sports fans) to the starter motor positive terminal.
Then, if the starter relay contacts have sufficient conductivity, 12 volts DC appear to the starter motor windings, and the starter motor, thus empowered, rotates and cranks the engine . . .
Now, putting aside the theoretical and going back to ground truth for a moment . . .
So, you turn on the ignition, engine in neutral (neutral light lit), kill switch in RUN position, press the starter button, and you only hear a click.
First field test, for starter relay and assorted connections, JUMP the great big connector on the starter relay to the tiny connector; does the starter motor then rotate? If so, your starter relay's o.k., and so is your starter. If not . . . you've either a faulty starter relay (interrupted solenoid wiring/bad contacts), or . . . your starter armature has a "dead spot" (a rotational orientation where no current flows in the armature), or bad brushes . . .
O.K., the jumped starter relay turns the engine over, let's say. Back up, and see if you get the + 12 volts out of the starter circuit relay when the ignition's on and the starter button is pushed . . .
If not, look back to the SAFETY SWITCH suite; are all these stars in alignment, as they should be?
Well, that's how I'd approach trouble-shooting the problem, using a multimeter and the wiring diagram (and maybe an alligator-clipped jumper wire or two).
If you get 12 VDC to the starter relay contol voltage terminal with the ignition key on and the starter button pressed, and still no cranking, the problem's in the starter relay or the starter itself. Jumping the starter relay (as described) wrings out the relay; still no crankie, the starter motor must be removed, disassembled and examined.
So, the chain of switches (e.g., ignition, starter button, safety switches) transfers the 12 VDC control voltage to the starter circuit relay; the starter circuit relay enables connection of 12 VDC to the starter relay control voltage terminal; the starter relay then connects the batter positive terminal through a heavy cable directly to the starter motor, and, one hopes, the motor "turns over."
Wish I could be ore specific, but . . . I'd be chasing wires with a multimeter and the wiring diagram myself, in your situation.