HIgh Street Kensington is fine if you're already on the Circle Line or District Line. However, if you're on the Piccadilly Line it's not worth changing; Gloucester Road is roughly the same distance from the RAH (as indeed is South Kensington), so get off there instead.
NB: It's about two thirds of a mile whichever station you use. So it's a brisk 10 minutes walk or nearer a quarter of an hour at a leisurely stroll. If you hate walking (or if the weather's bad), take a bus instead. Services 9, 10 and 52 pass the front of the RAH. Service 70 passes down the side of the RAH (along Queen's Gate) and service 360 terminates at the rear of the RAH (on Prince Consort Road).
Chris