I have a 2005 Harley Road King Custom. With ignition switch on, I have horn, brake lights and instrument light. I don't hear fuel pump, turn signals don't work; headlight doesn't work and

SOURCE: fuel pump?

the fuel pump for your 2000 fuel injected road king is located in the fuel tank. you will have to remove the speedometer housing, by first removing the seat, then remove the one small screw near the seat that holds the speedometer housing in place, also remove the small screw at the front of the speedometer housing and lift the housing off the tank and disconnect the wiring that goes to the back of the speedometer. there is a large oval shaped gold in appearance plate on top of the fuel tank. this plate is what the fuel pump is mounted to.

SOURCE: Speedo, turn signals, brake lights, horn stopped working. All else is fine.

Gap your plugs to 38 thou. Check all your bulbs (one burned out will affect the rest) Take a look at all your fuses and make sure they are all ok. Otherwise you are probably looking at a wiring short which will require trouble shooting of the elctrical system. Buy your self a service manual for your bike, this will help you a lot.

SOURCE: 1999 road king wont start

Its your fuel pump if you can't hear it. And its inside your tank. Not a hard fix if you have a book that tells you how to remove it. Alot of procedures to follow to get it right. And yes it does work off a relay.

SOURCE: lights, turn signals, horn don't work

Look for a bad ground connection inside the headlight shell (or possibly elsewhere). The other thing that all of the items you mention that are working intermittently have in common is the ignition switch. With the switch 'on', check for battery voltage at the fuses for the headlight, turn signals, horn, etc. No voltage at the fuses likely means a bad ignition switch.

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SOURCE: Brake lights,turn signal and horn not working...

Check the wiring within the headlight shell--look for connectors that may be corroded or pulled loose. A little WD-40 will favor connection integrity when sprayed into the connectors, then coupling and uncoupling them several times. Hope this helps!