Aberdeen International Airport is the closest airport to the castle. From the airport to the castle is a 39 minute drive.
Edinburgh Airport is also an option to fly in and out of. The drive from Edinburgh to the castle is 2 hours. If you choose to fly into Edinburgh, there is a train that will take you to Aberdeen city and then from there you can get a taxi or rent a car to drive to the castle.
Most flights from the US to Scotland have layovers in London. A third option for planning your trip would be to fly into London and from there take the express train or sleeper train to Aberdeen city.