How to become Private jet pilot?
Most corporate pilots don't get paid to fly, they get paid to wait. As Josh Martin and others have answered in great fashion, what you do with the waiting time is up to each individual pilot and dictated by whomever the employer/owner is and what their schedule entails.
Some corporate aviation missions are out at 8 am and back by 5-6 pm almost every day with predictable trips and leg lengths each day. Others get notified 1 hr in advance that they will be gone on a 2 week trip to BFE for 2 weeks. (Just in case you were wondering BFE is a real place in the middle of west Texas. Brownfield, TX to be exact and the airport identifier is KBFE. It is super hot in the summer and super cold in the winter with not much to do any time of the year and it is the epitome of the common use of the BFE acronym).
Flying a corporate jet is just like any other job in that their are good gigs to have and there are crappy ones. A lot depends on your qualifications, experience, etc, but a lot of corporate pilots start out much like any other job flying smaller aircraft for smaller companies and moving up to where they find the right fit as their qualifications improve.
Some pilots have dispatchers, hangar crews, maintenance crews, etc and they just show up at game time in order for the pre-game show in their slacks and dress shirt, say hi to the passengers as they get there and push the buttons and go. Others take care of every facet of the aircraft, including scheduling, maintenance (when appropriately licensed), do their own trip planning, working on the passenger's ground transportation needs, etc and are quite busy in between trips. Quite often, the more actual work your job entails is usually not commensurate to the pay you receive either, but that is part of the gig.
The bottom line is that private jet travel (and corporate/general aviation for that matter) is based about convenience and flexibility for the owners. It allows for things to get done both personally and professionally that are often times not possible without it. Before I go off on the tangent about private jet travel, I feel I must come back to the original scope of the question. The job of the pilot is to be flexible for the owners convenience, plain and simple. This usually means not having much control over your schedule, but making the best of being on the road.
Some pilots sleep, watch t.v., and do absolutely nothing when waiting. Others are productive, enjoy various sights and experiences they wouldn't get a chance to experience otherwise.
As I am writing this, I am in the Bay Area for a week on a trip for a client. The client offered to pay for my family to airline out here for the week (or airline me home for that matter if I wanted) as we have had a pretty aggressive flight schedule over the past 30 days and they want to make sure I am around the family as much as possible. We are getting a vacation and making the best of it. While I am here, I am also running my own company which involves buying and selling aircraft, managing multiple corporate aircraft for different companies, and a pool of corporate pilots.
Other folks are not so lucky. I have had those gigs too. I can honestly say that if I didn't enjoy it (as with most other pilots) we wouldn't be doing it. Flying the equipment we get to fly, going to the places we get to go are awesome. Feeling the kick in your pants and climbing at almost a mile a minute, working with/around the weather, overcoming challenges, and being the best aviation professional that I can be is what draws me to the job. Its a rush and a challenge and I actually quit another profession to do this one full time!