Well, there are a couple ways to go about it. The most obvious answer is to cure everybody in a timely fashion, but that's much more difficult than it implies. Patients will die if you don't cure them within a certain amount of time. You need very highly skilled staff, upgraded equipment, and a good hospital layout to ensure all your patients are taken care of quickly and properly.
This means having lots of GP offices; the core diagnosis loop is GP Office -> send patient for further diagnosis -> come back to GP Office -> diagnose illness or send back for further diagnosis. Rinse and repeat. So the GP Office is super important. As time goes on, they get really bogged down due to their heavy use.
You can trim this loop down by increasing your GP doctor's skills; there's a skill tree for General Practice, which ups diagnosis by 15% for doctors in GP offices, and specializing your GP doctors will ensure they are being very efficient. The faster they can diagnose illnesses, the less chance their patients will die before being cured.
Once you get the diagnosis loop as short as possible, you need to ensure your treatment skills are up to snuff; if not, you have a chance of killing your patients with the treatment. So you need nurses and doctors skilled in the Treatment branch; that ups their general treatment capability by 10% for each level, and your pharmacists should have the Pharmacy Management skill, too.
This is basically the strategy for the whole game. Not having anyone die is supposed to be the goal to strive for.
That said, there's a much simpler method to achieve this:
Just send them home when they're low on Health.