Becoming a better climber happens two ways. First, you need to spend more time working hill repeats after hill repeats. Second, you need to head to the gym and lift weights. Your power comes from your legs, so you will need to get especially comfortable with leg-focused exercises.
At the gym, stick to a set plan to get the most out of your workout. Maintain a log just like you do with your cycling. Keep your exercise selection small—four to five exercises are all you need—and stick to two days a week with no more than 45 to 60 minutes per session. Don't be shy with lifting heavy weights, either. You won't start there, but you can slowly build up to it by lifting slightly heavier weights each session.
Not sure where to start? The exercises below will give you the most bang for your buck.