I'm not a doctor so I've no idea what would fix this. In any case, I can do 30 push ups and 0 pull ups. And I practically never do bench press and train my back a lot more lol. These exercizes are more about bodyweight than anything.
First off, its not that your back is weak, you are weak EVERYWHERE.
rows and deadlifts will build your back a lot. overhead press alternated with bench press will build up your shoulders.
The general idea is to start with 'starting strength' or some '5x5 stronglifts beginner' program
Since the programs on their own say you should start with very low weight and add every time, they shouldn't hurt you. If you do one of them and start feeling pain in your back then make a post about it.(btw you WILL feel pain doing ANY decent program. Personally my knees hurt pretty much almost all the time. It's hard to try and figure out which one is/could lead to an injury and which one is just some passing pain that you should ignore. Maybe your pain is just some passing pain. I've had pains on my neck. Squatting fixed it.)