Technique 1 - Pedaling
My previous post "Pedalling - back to basics" covers various aspects of pedalling and much the same priciples apply when riding up hills although when climbing the cadence (how quickly the pedals are turned) would normally drop compared to when riding on the flat.
Many riders pedal too slowly up hills and rely on brute strength to keep moving perhaps moving body weight around to add some extra force to the pedals. Watching top riders climb you can see that most of them are actually still turning the pedals quite quickly and it is when a rider 'blows' that the cadence really drops.
How fast you should pedal would vary depending on the physical shape and natural rhythm of each rider but for a rider who rides at 90 to 100 rpm (revs per minute) on the flat would climb most hills with a cadence of 70 to 80 rpm.
For most riders that kind of pedalling rate seems very high and the only way to make it feel natural is to train at high cadence on hills, at first it will feel un-natural and you'll get breathless (see below for technique) more quickly but with perseverance it is effective.
The Science Bit
In simple terms when climbing at lower cadence the leg muscles will fatigue more quickly because of the higher force they have to apply to the pedals. At the same speed with a higher cadence the force exerted each time a pedal is pushed is lower BUT pedalling at a higher cadence will burn up more energy causing the heart and longs to work harder. There is a choice between putting more stress on your leg muscles or burning more energy. As long as energy supply and hydration are sorted the second is usually more effective.
There are couple of good articles on this subject see;
Technique 2 - Breathing
It seems almost too obvious to say, but it often needs to be said; However fit and strong you are if you don't breathe you won't move. If you don't breathe effectively then you won't climb effectively.
Ideally most breathing at areobic levels is done via the diaphragm and you should breathe in a controlled and calm manner - once you are gasping or taking short fast gulps of air then you are getting insufficient oxygen for the effort you are making, the work becomes anaerobic and you will fatigue quickly, power will drop and you will build up lactic acid which will hurt your muscles.
Technique 3 - Body Position
When climbing wind speeds are relatively low so win resistance is less important than it is on the flat. This allows you to ride in a more upright position and that in turn helps with effective breathing.
Keeping your head up will also ensure the air pathway to the lungs stays clear.
Sitting on the saddle when climbing uses fewer muscles than standing up so when climbing for a long time sitting down requires less oxygen. Sitting down and gripping the top of the bars (on dropped bars) reduces pressure on the diaphragm and keeps the airways open making breathing easier. The easier it is to breathe the more energy you can generate to climb. Often holding the bars toward the middle means that your elbows are further out from the body and this gives more room for the intercostal muscles (between the ribs) to move so that you can beathe more deeply.
Like the pedalling techniques above this does not always feel natural at first and repeated training using these techniques is required to make it feel more natural and to be able to climb like this for longer.
Sometime when you want to accelerate, the gradient is too steep or your gear not low enough (or you can't manage to maintain the seated position any longer) you will stand on the pedals - this allows you to apply more force to the pedals but the effort will tire leg muscles more quickly.
When climbing standing on the pedals (out of the saddle) you need to concentrate on breathing and balance. Breathing - keep your head up, keep elbows comfortably apart. Balance - keep enough weight on the back wheel to maintain good grip and prevent the wheel from slipping but keeping enough on the front wheel to keep it firmly on the road.
When moving from a sitting to standing position the bike will tend to slow down briefly while you make the transition and in close groups could cause a crash. To prevent that you should push harder on the pedals to lift yourself off the saddle rather than just standing up in a natural way.
Technique 4 - Timing and control of effort
Some riders climb best on short hills of a couple of minutes or less while others prefer long climbs taking to minutes or more. Whichever you prefer you are always going to have to climb the other sort of hills and you should measure your efforts so that you are riding strongly as you crest the summit. Knowing how hard to ride to crest the hill strongly comes with experience and getting to know where your own thresholds lie - both of those things come only with training - lots of training.
In summary - How to become a good climber
The old way to become a good climber was to get on your bike and climb hills, and do it again and again and again. That is still the best way but with attention to techniques and power to weight it will be even more effective.