I have always approached hill climbing the same way. (BTW, I am 205 lbs.) I focus on keeping my cadence consistant, and staying seated.

My training involves approaching the hill at a steady cadence, and shifting when my legs start to strain (before they are toast). I shift to maintain cadence regardless of the speed.

As I progress (this is good for repeating the same hill) I try to avoid shifting longer and longer.

You can use your shift points as a marker for improvement, and after a few cycles of this method, the hill gets easier and easier.