I too have dealt withlow back problems (sports injury), and as a medical student and patient I'll give you some insight as to what I did to help alleviate it. I would first go to a physician and see what exercises they say you can do. The two things that help it the most are weight lifting and yoga. I would avoid high impact running and hit the ellipticals (some of the higher end models at places like 24 hr fitness are pretty good at simulating running if thats your thing. What you should focus on for relieving back pain are your legs, gluts, abs (core) and back. Back extensions, as someone mentioned are a great way to strengthen your back, I would recommend doing these without weights. Gluts and core are the other two more important things to work on, yoga certainly helps. You can look up a variety of exercise regimens and see what are the best for you. Don't neglect your upper back and shoulders. Remember not to overdo it for the first month to 6 weeks. Lastly Legs outta be a focal point as well. As you strengthen the core and lower back muscles (along with gluts/legs/upper back), you outta notice your back pain go away. Stay away from some of the explosion type exercises (power cleans, others) until you have a significant change in back pain, even then I would go to a physician and make sure they approve. As an alternative I hear P-90X is really good, this type of combined aerobic anaerobic exercise is the new wave of exercise and really helps shave off the lbs pretty quickly
As far as what will help you hit the ball better. The muscles involved are the "inside muscles" of the leg (adducters, gluts), the hamstrings, hips, medial back muscles, shoulder (including rotater cuff), and your forearms. Working out theoretically all your muscles should help your golf game but getting rid of that back pain would really help you out. I found what helped me most was working my forearms. There is a 20 dollar gripper device called the captains of crush. This device will signficantly increased your forearm strength. You can look up a a variety of exercises for the other muscles though.