Find a company that flies private planes to Bermuda from wherever you live. Check out Air Broker Net, Private Air Charter or Air Charter Guide which all fly to Bermuda. Air Charter Guide also offers a quick quoting service on their website to give you an estimate of how much your trip to Bermuda will cost.
Make sure the plane company is Part 135 certified by the Federal Aviation Administration. Part 135 certified means that the FAA has made sure this company meets all requirements in order to fly passengers. Do not fly with a private company that isn't Part 135 certified. If you're unsure, call the FAA office in your area, which you can find on their website.
Figure out how many people are coming with you and how much baggage each person is bringing. This will help you figure out what kind of private plane you need to charter. The more people that pitch in, the cheaper it will be for everybody. Always try to fill the private plane you charter to capacity.
Call each company to get a quote on price. You'll often be charged an hourly rate or rate per mile, a landing fee and possible customs or international charges. Watch out for overnight fees if you need to pay for the crew to stay in Bermuda, or a deadhead fee, which is when the flight needs to return without passengers. Ask the company if you'll be charged this fee.
Book the time with the private plane company you chose, and ask for the exact total price you'll be charged, sometimes called the hard quote. Make sure there are no hidden fees, that all your passengers are coming and that your luggage won't be too heavy for the plane.