The best thing to clean a bathtub with is some vinegar and dish soap or even some toilet bowl cleaner or Ajax cleaner.
I've had good luck with cleaning a really griming bathtub with some toilet bowl cleaner and normal grime I get rid of with some Ajax cleaner and even dish soap and vinegar works wonders at cleaning most bathtub grime.
To clean a bathtub without hurting your back you can use a good sponge mop to clean the bathtub with like you would with a normal sponge but the handle will prevent you from having to bend down and hurt your back.
A clean angle broom with synthetic bristles is also a good cleaning tool to clean a bathtub without bending down.
You can also use some toilet bowl cleaner to clean really stubborn grime from a bathtub.
I've also used a mop to clean my bathtub when I didn't really need to scrub it down.
You can also put a scrub brush on a long pole or stick to increase the length of the handle so you won't have to bend down but a good angle brush works wonders at cleaning bathtubs without needing to bend down.
Another good cleaning solution for bathtubs is a mixture of white vinegar, essential oils and dish soap or even just white vinegar and dish soap or dish soap alone.
If your bathtub is especially grimy, combine equal parts white vinegar and dish soap in a glass measuring cup.
Microwave it until hot (about 30 seconds), then stir.
Pour the solution all over the tub, let it sit for 30 minutes, then sweep.
The shower head can be used to rinse the bathtub out after scrubbing and if you have a handheld shower head it will work better at rinsing the the bathtub.
Using a decent good sponge mop works well for cleaning a bathtub or shower when you have trouble bending over and scrubbing with a normal sponge.
I've used a good sponge mop for cleaning my tub and shower with and it has helped tremendously and has saved by back.