Hi Alix,

I would look at some of the more obvious answers first...changing out your running shoes, considering inserts and so on.  It may be due to muscle strain, but also due to leg length difference, hip discrepency, pressure on spinal discs or basically a combination of things that deal with distribution and force of your weight while running.  Most likely, a foot imbalance is causing the problem, but it is important to check out all the possibilities.  Consider tapering down your training schedule and avoid hills until you have a professional opinion.  Also, swimming and cycling might be good cross-training alternatives if it is recommended to reduce the impact on your lower back. 

Hope you are feeling better soon!

You need to be doing a lot of core and lower back exercises, running puts a lot of strain on your lower back.  A strong core will help you a ton!  make sure you are running tall, proper form.