At 5'11" and 232 lbs (and dropping fast), I'm certainly not the picture of the typical cyclist's frame. That said though, I've found that riding about the same distances as you over the summer, I've greatly built up my strength and speed, especially on climbs. I've been climbing the Strava Leaderboards on local climbing segments, and am generally in the top 10% overall on such segments now.

Here's what I've noticed: I commute regularly to work. I have a ride with about 600' of total ascent going in, and about 900' coming home. One day I decided to do some hill repeats on hills between my house and the river (I'm at about 330' elevation, and river is essentially sea level here). The next day on my commute, I was setting personal records on steep hills, and noticed I was maintaining my cadence through the longer, shallower inclines without really having to downshift. Immediate effect.

The following weekend, the wife and I rode our kids to the pool. I towed them in our bike trailer, which weighs over 100 lbs once you load it with the kids. Only put on about 200' of elevation going to the pool and about 150' coming home, but again the rest of the week I just felt WAY stronger.

Anyways, while this is by no means scientific, I've noticed that anything you do to build up your leg strength will likely translate into noticeable improvement in your climbing. Do the hill repeats, or borrow a trailer and load it down, or just do some squats/leg presses/whatever to build up your strength.

Also, if you're doing hill repeats, I'd recommend using an app like Strava. You can search for good hills in your area (they'll be marked with segments that you can compete on). Also, the segment leaderboards will a) let you track your improvement over time, and b) push you to compete with others.