Any skater who is serious enough to retain a coach to train for competitions quickly recognizes the importance of off-ice training. Besides ballet, USFSA also recommends strength training; see their general guidelines for strength training here.
Boring as heck, three strength exercises are essential to getting to the next level: sit ups, pull ups, and shoulder presses. Sit ups provide the strength to keep a straight core and properly aligned spine while you spin. Shoulder presses allow you to maintain control over your arm movements.
Everyone though seems to overlook pull ups. The back muscles strengthened by pull ups do more for your posture than any other muscle group.
Once you move up to elite you likely need a more serious and educated personal trainer. See for example this article by Charles Poliquin of how he has trained world-class Canadian skaters.
Some basic strength training at the rink or at the gym is infinitely helpful to your competitive skating, but once you get committed at a national level be sure to kick your off-ice training up to the next level.