Now I am not a pro, nor a sports physician, but I do think analysis of pro cyclists data can give you a good insight in how to train. First of all, and therewith risking to state the obvious, Merckx expressed it most eloquently, namely that you get good at what you do a lot. If you want to be a good climber, climbing hills or mountains is the best way to go.
Secondly and more concretely, I think interval trainings are most beneficial, as they give quick results in a relatively low time-consuming way. The problem then is, what kind of intervals should you do?
If you do have hills or mountains in your area, it's fairly easy, so I would suggest doing intervals (3 reps of 6 mins to 6x6 on / every rep followed by an off dependent on your fitness level) 2 to 3 times a week. Cadence around 60-70 and heart rate 95%.
On the other hand, most of us don't live near mountains or long enough hills so you will want to simulate climbing conditions throughout your intervals. During wintertime you can do this wondefully on your indoor trainer. I'd recommend doing 3x6 to 6x6 mins on / off depending on your fitness level, on high gear and low cadence. You need to push hard similar to what you would feel doing high rep leg presses to rotate and get a high heart rate. Again aim for low cadence 60-70 and high heart rate.
I think the following charts are pretty instructive, mostly because they are are from one of the best young climbers in the peloton.