How much electrical power is available in a Gulfstream G550 and from which power sources?
There are some answers to common electrical questions about the G550 here according to them the engine integrated drive generator (IDG) unit delivers 40KVA, 400HZ, 115V; they don't list a power factor but lets be conservative and assume .8 (it must be between 0 and 1). This would result in 32 Kilowatts per IDG. So you have 64 Kilowatts at your disposal. The minimum load will depends on the avionics installed and can vary from plane to plane. At night load will be increased as a result of the use of interior lights as well as landing lights during takeoff and approach. Your load will be highly variant on the interior of the aircraft. As this is a luxury plane there could be a whole host of things inside like TV's, lights, kitchen equipment etc. You can find specific equipment installations in the POH/loading info for a given airframe. You can find a nice breakdown of the G450's electrical system here.  

As for how you can add it, you can simply add it as you would add any accessory, to the main bus (by an A&amp;P of course). You will need to make sure the item can be added to the craft and in some cases there may be a stipulation along the lines of not using it during takeoff or landing. The amount of load will vary by aircraft and dependent on what else is installed.   


  I'd like to understand what electrical power sources are available
  (engines? APU? GPU? Other?)


The engines (while running) are generally the main source of power. The APU is often used on the ground when the engines are off but the plane needs electronics on and can also serve to help start the engines. It does appear the G550 has an APU. According to this data sheet the G550 runs a Honeywell RE220 APU that can output 45 Kilowatts. According to the site it will operate up to 37,000+ Ft. so it in theory could be run for the duration of a flight. 

A GPU is only available on the ground and only at airports that have them. Their output varies widely and a large variety of types may be available at any given airport.