Back pain when cycling up hills
There are so many factors involved that can lead to back pain that I would be suspicious of anyone providing the answer.

Some factors that can affect 

Your current bike fit.


Body angle: is your current fit very aggressive or more relaxed?
Is you seat set back appropriate or too far back?
Is your saddle height correct for your level of flexibility?


Your Technique


How do you position yourself on your bike  when climbing  (e.g., do you use the tops of the bars or do you get in the drops)?
Do you spin or do you mash gears


Your current bike setup


What gearing do you currently have
Is this gearing sufficiently low enough for climbs


Other personal factors


Have you injured your back before
What is your hamstring flexibility
What is your mid and lower back flexibility
What is your hip flexors flexibility
The list goes on


Summary

After addressing all of these issues I would then start looking at factors such as core strength, functional weaknesses and associated exercises. 

In short, narrowing down the cause of back pain is often can be difficult.  I would suggest finding a someone who specializes in both physio and bike fitting. They exist and will likely be your best bet at tracking down the cause and finding a solution.