The license to fly a bizjet is the same as flying an airliner. To be a captain of a bizjet (most of them any way) you'll need an ATPL of some sorts. JAA, FAA whatever. The license required is dependent on where the aircraft is registered.
A pilot's license doesn't have any thing to do with immigration. If you want to work in the US you'll need the appropriate visa/green card.
How people get jobs on bizjets is varied. Some, like Netjets do an interview not unlike an airline style interview, others you get the job because you're known to the operator and they like the look of you.
In other words, networking.
it's fun and very interesting flying.